Projective Coordinates for Points and Lines

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<div id="description" class="style-scope ytd-video-secondary-info-renderer">Dr Anna Tomskova explains a more modern framework for projective geometry where the extra coordinate often associated with infinity is the first coordinate in a projective vector. This gives us a uniform way to associate to affine points and lines projective points and lines, with the advantage that now higher dimensional vectors become extensions, not rewrites of lower dimensional vectors, and equations of lines, and eventually also curves, respect the natural increasing order of degree. The Join of Points and Meet of Lines formulas, which are the workhorses for planar analytic geometry, are pleasantly symmetrical, and the dual role of points and lines should be clear. This is an important computational tool for the Algebraic Calculus, as it generally allows us to clear denominators more efficiently, meaning that rational number arithmetic is reduced to integer arithmetic --which we generally all prefer! Also this gives us a simple algebraic way of talking about points at infinity and the line at infinity. This video is a reworking of an earlier video in the series.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">so now we are going to introduce the</div>

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<div class="cue style-scope ytd-transcript-body-renderer">idea of projective coordinates for</div>

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<div class="cue style-scope ytd-transcript-body-renderer">points and lines we already know what</div>

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<div class="cue style-scope ytd-transcript-body-renderer">points and lines are now we want</div>

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<div class="cue style-scope ytd-transcript-body-renderer">alternative way of representing them the</div>

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<div class="cue style-scope ytd-transcript-body-renderer">way that actually brings them more</div>

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<div class="cue style-scope ytd-transcript-body-renderer">closely in line with each other so that</div>

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<div class="cue style-scope ytd-transcript-body-renderer">points and ones and start to look quite</div>

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<div class="cue style-scope ytd-transcript-body-renderer">similar and the statements we'll make</div>

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<div class="cue style-scope ytd-transcript-body-renderer">about them become much more symmetric</div>

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<div class="cue style-scope ytd-transcript-body-renderer">there are a lot of odd advantages to</div>

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<div class="cue style-scope ytd-transcript-body-renderer">this projective somewhat larger point of</div>

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<div class="cue style-scope ytd-transcript-body-renderer">view including the fact that allows</div>

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<div class="cue style-scope ytd-transcript-body-renderer">asked to work with fractions a little</div>

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<div class="cue style-scope ytd-transcript-body-renderer">bit more smoothly actually allows us to</div>

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<div class="cue style-scope ytd-transcript-body-renderer">remove denominators more efficiently so</div>

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<div class="cue style-scope ytd-transcript-body-renderer">replace fractional thinking with integer</div>

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<div class="cue style-scope ytd-transcript-body-renderer">thinking and it's also turn out to have</div>

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<div class="cue style-scope ytd-transcript-body-renderer">advantages when we work with a point at</div>

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<div class="cue style-scope ytd-transcript-body-renderer">infinity and sort of perspective but</div>

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<div class="cue style-scope ytd-transcript-body-renderer">it's not gonna play a major role for us</div>

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<div class="cue style-scope ytd-transcript-body-renderer">at least initially maybe is that there</div>

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<div class="cue style-scope ytd-transcript-body-renderer">is a fine geometry and the risk</div>

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<div class="cue style-scope ytd-transcript-body-renderer">projective geometry without worrying too</div>

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<div class="cue style-scope ytd-transcript-body-renderer">much about what projective geometry is</div>

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<div class="cue style-scope ytd-transcript-body-renderer">by going to take just some key ideas</div>

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<div class="cue style-scope ytd-transcript-body-renderer">from it so what we're gonna do we are</div>

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<div class="cue style-scope ytd-transcript-body-renderer">going to take two ideas that we</div>

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<div class="cue style-scope ytd-transcript-body-renderer">currently have of points and lines in</div>

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<div class="cue style-scope ytd-transcript-body-renderer">the fine geometry and look at the</div>

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<div class="cue style-scope ytd-transcript-body-renderer">corresponding way of representing those</div>

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<div class="cue style-scope ytd-transcript-body-renderer">objects in projective geometry so how</div>

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<div class="cue style-scope ytd-transcript-body-renderer">does it go if you have a point a which</div>

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<div class="cue style-scope ytd-transcript-body-renderer">is X Y in the fine geometry we can write</div>

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<div class="cue style-scope ytd-transcript-body-renderer">it as a projective point by considering</div>

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<div class="cue style-scope ytd-transcript-body-renderer">a proportion 1/2 X 2y enclosed in square</div>

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<div class="cue style-scope ytd-transcript-body-renderer">brackets and square brackets are still</div>

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<div class="cue style-scope ytd-transcript-body-renderer">alerting us that we have a point but</div>

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<div class="cue style-scope ytd-transcript-body-renderer">instead of having two numbers separated</div>

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<div class="cue style-scope ytd-transcript-body-renderer">by a comma we have three numbers which</div>

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<div class="cue style-scope ytd-transcript-body-renderer">are forming</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Oshin 1 2 X 2 1 now what about the line</div>

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<div class="cue style-scope ytd-transcript-body-renderer">his line L in equation form which we</div>

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<div class="cue style-scope ytd-transcript-body-renderer">have been introduced it as an equation</div>

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<div class="cue style-scope ytd-transcript-body-renderer">ax plus B y equals C and we have said</div>

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<div class="cue style-scope ytd-transcript-body-renderer">that this equation is going to be</div>

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<div class="cue style-scope ytd-transcript-body-renderer">unchanged when we say multiply all of</div>

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<div class="cue style-scope ytd-transcript-body-renderer">the entries by some number it's very</div>

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<div class="cue style-scope ytd-transcript-body-renderer">suggestive of having a proportion in</div>

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<div class="cue style-scope ytd-transcript-body-renderer">fact and suggest to us that we should</div>

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<div class="cue style-scope ytd-transcript-body-renderer">really be thinking about proportion of</div>

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<div class="cue style-scope ytd-transcript-body-renderer">these three numbers rather than those</div>

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<div class="cue style-scope ytd-transcript-body-renderer">three numbers themselves so we are going</div>

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<div class="cue style-scope ytd-transcript-body-renderer">going to do a slight twist we're gonna</div>

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<div class="cue style-scope ytd-transcript-body-renderer">think about this equation as minus C</div>

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<div class="cue style-scope ytd-transcript-body-renderer">plus X was before we proceed please pay</div>

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<div class="cue style-scope ytd-transcript-body-renderer">attention to the order of the terms here</div>

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<div class="cue style-scope ytd-transcript-body-renderer">in this course it will be an agreement</div>

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<div class="cue style-scope ytd-transcript-body-renderer">that we write the constant term first</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and the other terms in such order that</div>

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<div class="cue style-scope ytd-transcript-body-renderer">the power of variables are increasing</div>

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<div class="cue style-scope ytd-transcript-body-renderer">you will feel convenience of</div>

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<div class="cue style-scope ytd-transcript-body-renderer">disagreement when we have quadratic</div>

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<div class="cue style-scope ytd-transcript-body-renderer">cubic expressions high degree</div>

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<div class="cue style-scope ytd-transcript-body-renderer">polynomials and even series in this</div>

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<div class="cue style-scope ytd-transcript-body-renderer">course so if we think about this that</div>

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<div class="cue style-scope ytd-transcript-body-renderer">way we can encode that button by these</div>

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<div class="cue style-scope ytd-transcript-body-renderer">three coefficients - C - a - B and that</div>

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<div class="cue style-scope ytd-transcript-body-renderer">is now proportion and we in causing that</div>

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<div class="cue style-scope ytd-transcript-body-renderer">proportion in round brackets to alert us</div>

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<div class="cue style-scope ytd-transcript-body-renderer">the point points in square brackets</div>

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<div class="cue style-scope ytd-transcript-body-renderer">lines in normal rackets okay</div>

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<div class="cue style-scope ytd-transcript-body-renderer">let's give an example this is a</div>

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<div class="cue style-scope ytd-transcript-body-renderer">classical a fine point one salt for fish</div>

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<div class="cue style-scope ytd-transcript-body-renderer">according to this we can rewrite it as a</div>

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<div class="cue style-scope ytd-transcript-body-renderer">projective coordinate as 1/2 1/3 - for</div>

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<div class="cue style-scope ytd-transcript-body-renderer">fish we have made agreement that these</div>

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<div class="cue style-scope ytd-transcript-body-renderer">things equal to each other this is the</div>

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<div class="cue style-scope ytd-transcript-body-renderer">convention that we adopt in</div>

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<div class="cue style-scope ytd-transcript-body-renderer">but now because this is a proportion we</div>

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<div class="cue style-scope ytd-transcript-body-renderer">are allowed to multiply the numbers here</div>

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<div class="cue style-scope ytd-transcript-body-renderer">but anything that we wanted and the</div>

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<div class="cue style-scope ytd-transcript-body-renderer">proportion is not going to change in</div>

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<div class="cue style-scope ytd-transcript-body-renderer">particular we can multiply it by this</div>

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<div class="cue style-scope ytd-transcript-body-renderer">common multiple of the denominators here</div>

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<div class="cue style-scope ytd-transcript-body-renderer">so we can multiply by 15 essentially</div>

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<div class="cue style-scope ytd-transcript-body-renderer">clean clearing these fractions so we</div>

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<div class="cue style-scope ytd-transcript-body-renderer">multiply oil all three entries by 15</div>

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<div class="cue style-scope ytd-transcript-body-renderer">we're gonna get 15 to 5 to 12 so this</div>

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<div class="cue style-scope ytd-transcript-body-renderer">first course in it that we introducing</div>

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<div class="cue style-scope ytd-transcript-body-renderer">which you may think is kind of waste of</div>

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<div class="cue style-scope ytd-transcript-body-renderer">time actually performs the useful role</div>

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<div class="cue style-scope ytd-transcript-body-renderer">that it's a place where we can create</div>

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<div class="cue style-scope ytd-transcript-body-renderer">the nominators form so starting with</div>

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<div class="cue style-scope ytd-transcript-body-renderer">rational numbers multiplying by the</div>

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<div class="cue style-scope ytd-transcript-body-renderer">least common multiple of the</div>

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<div class="cue style-scope ytd-transcript-body-renderer">denominators we get three integers this</div>

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<div class="cue style-scope ytd-transcript-body-renderer">is a powerful computational device</div>

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<div class="cue style-scope ytd-transcript-body-renderer">because most of us prefer to work with</div>

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<div class="cue style-scope ytd-transcript-body-renderer">integers rather than with fractions but</div>

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<div class="cue style-scope ytd-transcript-body-renderer">they still contain the same amount of</div>

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<div class="cue style-scope ytd-transcript-body-renderer">information if you want to go backwards</div>

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<div class="cue style-scope ytd-transcript-body-renderer">from the proportional point of view</div>

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<div class="cue style-scope ytd-transcript-body-renderer">toward the point that we have just to do</div>

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<div class="cue style-scope ytd-transcript-body-renderer">is divide by this first entry then it</div>

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<div class="cue style-scope ytd-transcript-body-renderer">becomes one and these two become these</div>

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<div class="cue style-scope ytd-transcript-body-renderer">two fractions which determine the point</div>

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<div class="cue style-scope ytd-transcript-body-renderer">let's have a look at an example of a</div>

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<div class="cue style-scope ytd-transcript-body-renderer">line but line L which is minus 7x plus</div>

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<div class="cue style-scope ytd-transcript-body-renderer">the y equals 4 well if I bring that for</div>

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<div class="cue style-scope ytd-transcript-body-renderer">to the other side it becomes minus 4 and</div>

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<div class="cue style-scope ytd-transcript-body-renderer">then we can represent this line by the</div>

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<div class="cue style-scope ytd-transcript-body-renderer">proportions - for two - 7 - 2 all in in</div>

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<div class="cue style-scope ytd-transcript-body-renderer">round brackets alerting us to the fact</div>

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<div class="cue style-scope ytd-transcript-body-renderer">that this is a line not a point and not</div>

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<div class="cue style-scope ytd-transcript-body-renderer">once we have done this we could if we</div>

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<div class="cue style-scope ytd-transcript-body-renderer">don't like it multiply this proportion</div>

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<div class="cue style-scope ytd-transcript-body-renderer">number that we want including -1 I want</div>

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<div class="cue style-scope ytd-transcript-body-renderer">to rewrite it the hub slightly in a more</div>

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<div class="cue style-scope ytd-transcript-body-renderer">nice await yes 4 - 7 - - - there is</div>

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<div class="cue style-scope ytd-transcript-body-renderer">another example of a lion M which is in</div>

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<div class="cue style-scope ytd-transcript-body-renderer">the form y equals 3 over 2x minus 5 this</div>

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<div class="cue style-scope ytd-transcript-body-renderer">is not exactly the standard form that</div>

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<div class="cue style-scope ytd-transcript-body-renderer">the same way we had here so let's think</div>

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<div class="cue style-scope ytd-transcript-body-renderer">about how it we can rewrite this in</div>

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<div class="cue style-scope ytd-transcript-body-renderer">standard form</div>

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<div class="cue style-scope ytd-transcript-body-renderer">well look at we can multiply everything</div>

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<div class="cue style-scope ytd-transcript-body-renderer">by 2 to get rid of this denominator to</div>

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<div class="cue style-scope ytd-transcript-body-renderer">get 2y equals 3x minus 5 and then we</div>

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<div class="cue style-scope ytd-transcript-body-renderer">bring everything to one side to get</div>

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<div class="cue style-scope ytd-transcript-body-renderer">minus 10 plus 3x minus 2y once I have</div>

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<div class="cue style-scope ytd-transcript-body-renderer">this form we can immediately write down</div>

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<div class="cue style-scope ytd-transcript-body-renderer">what is projective form of this line is</div>

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<div class="cue style-scope ytd-transcript-body-renderer">it is three coefficients the constant</div>

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<div class="cue style-scope ytd-transcript-body-renderer">term is minus 10 the coefficient of x is</div>

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<div class="cue style-scope ytd-transcript-body-renderer">3 and the coefficient of Y is minus 2 I</div>

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<div class="cue style-scope ytd-transcript-body-renderer">hope you can appreciate it that there is</div>

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<div class="cue style-scope ytd-transcript-body-renderer">something powerful about this point of</div>

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<div class="cue style-scope ytd-transcript-body-renderer">view because when we are talking about</div>

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<div class="cue style-scope ytd-transcript-body-renderer">equation of Y it was a little bit</div>

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<div class="cue style-scope ytd-transcript-body-renderer">ambiguous right um there are all these</div>

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<div class="cue style-scope ytd-transcript-body-renderer">different possible ways of rewriting</div>

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<div class="cue style-scope ytd-transcript-body-renderer">this equation</div>

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<div class="cue style-scope ytd-transcript-body-renderer">maybe with wild left hand side maybe it</div>

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<div class="cue style-scope ytd-transcript-body-renderer">when we convert to this projective form</div>

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<div class="cue style-scope ytd-transcript-body-renderer">then these ambiguities mostly disappear</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and the only ambiguous that is left is</div>

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<div class="cue style-scope ytd-transcript-body-renderer">the fact that we can multiply all three</div>

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<div class="cue style-scope ytd-transcript-body-renderer">numbers by scalar</div>

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<div class="cue style-scope ytd-transcript-body-renderer">zero scalar so conceptually this is</div>

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<div class="cue style-scope ytd-transcript-body-renderer">actually very nice an interactive way of</div>

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<div class="cue style-scope ytd-transcript-body-renderer">thinking about points what points and</div>

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<div class="cue style-scope ytd-transcript-body-renderer">lines are but have started by defining a</div>

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<div class="cue style-scope ytd-transcript-body-renderer">fine points and lines in this way but we</div>

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<div class="cue style-scope ytd-transcript-body-renderer">prefer to think about points and lines</div>

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<div class="cue style-scope ytd-transcript-body-renderer">in this more traditional way and just</div>

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<div class="cue style-scope ytd-transcript-body-renderer">think about this as being alternative</div>

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<div class="cue style-scope ytd-transcript-body-renderer">way of representing them so this are the</div>

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<div class="cue style-scope ytd-transcript-body-renderer">projective ways of representing points</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and lines okay so here we come to</div>

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<div class="cue style-scope ytd-transcript-body-renderer">important formula called the projective</div>

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<div class="cue style-scope ytd-transcript-body-renderer">incidence formula from our point of view</div>

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<div class="cue style-scope ytd-transcript-body-renderer">we are very interested in formulas this</div>

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<div class="cue style-scope ytd-transcript-body-renderer">course will be full of interesting</div>

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<div class="cue style-scope ytd-transcript-body-renderer">formulas and think of formulas as being</div>

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<div class="cue style-scope ytd-transcript-body-renderer">just as important as theorems generally</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and if we can we want to present</div>

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<div class="cue style-scope ytd-transcript-body-renderer">information in the form of formulas we</div>

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<div class="cue style-scope ytd-transcript-body-renderer">feel a little bit more comfortable and</div>

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<div class="cue style-scope ytd-transcript-body-renderer">confident about validating and</div>

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<div class="cue style-scope ytd-transcript-body-renderer">interpretating and verifying formulas</div>

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<div class="cue style-scope ytd-transcript-body-renderer">okay so this is the way of thinking</div>

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<div class="cue style-scope ytd-transcript-body-renderer">about incidence relationship between</div>

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<div class="cue style-scope ytd-transcript-body-renderer">points and Wiles in the language of</div>

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<div class="cue style-scope ytd-transcript-body-renderer">projective coordinates and the statement</div>

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<div class="cue style-scope ytd-transcript-body-renderer">is that the point a let's say A to B to</div>

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<div class="cue style-scope ytd-transcript-body-renderer">C if that's the projective form of the</div>

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<div class="cue style-scope ytd-transcript-body-renderer">point is incident with the line L which</div>

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<div class="cue style-scope ytd-transcript-body-renderer">is R - s - t that they project the form</div>

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<div class="cue style-scope ytd-transcript-body-renderer">of the line precisely when this relation</div>

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<div class="cue style-scope ytd-transcript-body-renderer">is satisfied that this a times R plus B</div>

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<div class="cue style-scope ytd-transcript-body-renderer">times s plus C times T equals zero in</div>

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<div class="cue style-scope ytd-transcript-body-renderer">other words if you take definition of</div>

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<div class="cue style-scope ytd-transcript-body-renderer">incidence that originally have in a fine</div>

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<div class="cue style-scope ytd-transcript-body-renderer">case and convert it into this projective</div>

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<div class="cue style-scope ytd-transcript-body-renderer">setting then this is the</div>

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<div class="cue style-scope ytd-transcript-body-renderer">test about whether this point and this</div>

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<div class="cue style-scope ytd-transcript-body-renderer">line are incident so for example we can</div>

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<div class="cue style-scope ytd-transcript-body-renderer">check that the point 4 - realize on the</div>

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<div class="cue style-scope ytd-transcript-body-renderer">line 3x plus 2y equals 6 we can do it</div>

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<div class="cue style-scope ytd-transcript-body-renderer">directly just but substituting this in</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and see if it works</div>

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<div class="cue style-scope ytd-transcript-body-renderer">3 times 4 plus 2 times minus 3 yes</div>

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<div class="cue style-scope ytd-transcript-body-renderer">that's actually equal 6 so yes this</div>

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<div class="cue style-scope ytd-transcript-body-renderer">point does lie on the line this line but</div>

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<div class="cue style-scope ytd-transcript-body-renderer">now we can also do this using projective</div>

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<div class="cue style-scope ytd-transcript-body-renderer">coordinates let's see how that would</div>

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<div class="cue style-scope ytd-transcript-body-renderer">work so this point would be represented</div>

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<div class="cue style-scope ytd-transcript-body-renderer">protectively as the point 1 2 4 2 minus</div>

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<div class="cue style-scope ytd-transcript-body-renderer">3 that's how we go from point in the</div>

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<div class="cue style-scope ytd-transcript-body-renderer">final coordinates to projective</div>

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<div class="cue style-scope ytd-transcript-body-renderer">coordinates this line to be represented</div>

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<div class="cue style-scope ytd-transcript-body-renderer">by the projective coordinates - 6 - 3 -</div>

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<div class="cue style-scope ytd-transcript-body-renderer">2 okay and then we can check incidence</div>

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<div class="cue style-scope ytd-transcript-body-renderer">just by playing this rule 1 times minus</div>

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<div class="cue style-scope ytd-transcript-body-renderer">6 plus 4 times 3 plus -3 times 2 equals</div>

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<div class="cue style-scope ytd-transcript-body-renderer">that's minus 6 that's 12 it's minus 6 so</div>

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<div class="cue style-scope ytd-transcript-body-renderer">yes forget 0 we get 0 so yes this point</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and this line are incident let's how we</div>

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<div class="cue style-scope ytd-transcript-body-renderer">check that relation in projective</div>

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<div class="cue style-scope ytd-transcript-body-renderer">coordinates so next I'm going to show</div>

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<div class="cue style-scope ytd-transcript-body-renderer">you two very important formulas that</div>

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<div class="cue style-scope ytd-transcript-body-renderer">allows us to do two fundamental things</div>

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<div class="cue style-scope ytd-transcript-body-renderer">we want to do first of all just to find</div>

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<div class="cue style-scope ytd-transcript-body-renderer">the join of two points you have two</div>

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<div class="cue style-scope ytd-transcript-body-renderer">points how to find the equation of a</div>

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<div class="cue style-scope ytd-transcript-body-renderer">line let's join them and then secondly</div>

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<div class="cue style-scope ytd-transcript-body-renderer">how to find the mid of two lines I'm</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">going to see that this projective point</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">of view really make these problems very</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">interesting and attractive ok so the</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">giant of the points in beam and the</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">point a is a to be to see and avoid B is</div>

</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">this is the following line L which we're</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">gonna first of all write us a B so it B</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">means the line that joins a be the line</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">that passes through a and B and he is</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">the formula for it projective coordinate</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">wise so you see this by the sound</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">records and run three entries and then</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">enters a what</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">the entries are the F minus ze that's</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">the first entry in the second entry is C</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">D minus AF and the third entry is a II</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">minus BD that might look like or rather</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">imposing formula but I do remind you</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">that those like exactly the kind of</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">quantities that appeared when we were</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">investigating the question when a given</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">three proportion equals another three</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">proportion so this thing here is just</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">the second and third entries here here</div>

</div>

</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">B times F minus C times m and the second</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">entry just involves the first and third</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">entry of bosses so this is C times D</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">minus a times F and the third entry just</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">involves the two first entries here so</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">this is a times e minus B times D so for</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">example if a is a point 3 1 and B is a</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">point minus 1/4 is the diagram there's a</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">point eight three one</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Andrew point B which is minus 1/4 then</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">we might want to find the equation of</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">the line L passes through these two</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">points</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">in high school you may have learned that</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">in order to solve this you need to solve</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">some system of two linear equations this</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">is the alternate way</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">and very powerful way of solving this</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">important problem what we do we convert</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">the points into projective coordinates</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">so 3 1 becomes 1 2 3 2 1 and minus 1 4</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">becomes 1 2 minus 1 2 4 and then we just</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">apply this formula to write down</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">immediately</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">the projective equation of the line</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">passes so these two points which is what</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">okay so the first entry is obtained by</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">looking at the second and third entries</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">here so 3 times 4 minus 1 times minus 1</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">the same second entry is obtained by</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">looking at the first and third entries</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">here it's 1 times 1 minus 1 times 4 and</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">the third entry is obtained by looking</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">at the first two entries here it's 1</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">times minus 1 minus 3 times 1 and that</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">amounts to 3 times 4 minus 1 is minus 1</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">this is our 12 plus 1 is 13 the second</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">entry is 1 times 1 minus 4 it's minus 3</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">and 1 times minus 1 minus 3 is minus 4</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">so we get to sit in 2 minus 3 to minus 4</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">now if we can multiply this proportion</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">by minus 1 we get minus 14 to 3 to 4 and</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">then the equation of this line must be</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">minus 13 plus 3x plus 4y or this is 4x</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">plus 4y equals 13 just to check that</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">let's just check that the point 3 1</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">satisfy this line this is 3 times 3</div>

</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">we plug in minus 1 4 minus 1 here for</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">the air we get minus 3 plus 16 yes</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">that's 13 as well so this is indeed the</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">equation of the line that passes through</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">those two points</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">so we can solve the bench of high school</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">algebra problems by just remembering</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">this particular formula here it's very</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">important formula involved how to fire</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">the joint of the points in projective</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">coordinates so there is a dual problem</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">the very similar kind of problem to ask</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">if you have two lines say Alan M how to</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">find the point which lies on both of</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">them so this is of meat/meat of lines</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">formula the middle lines L let's say the</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">line is a to B to C in projective</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">coordinates and M which is say D to e to</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">F also in projective coordinates is the</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">point a and B right LM to represent that</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">this is the mid of two lines LM and is</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">given by this formula in square brackets</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">and there are three qualities and these</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">quantities are exactly the same</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">qualities as appeared and previous</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">follower so if you memorize previous</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">formula you automatically know this one</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">just remember that output is the point</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">therefore we divide in square brackets</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">for example here to line that we looked</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">at earlier the one L which is 3x plus 2y</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">equals 6</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">and the line m with the equation X minus</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">y equals 1 and we'll be interested in</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">where these two lines meet in projective</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">coordinates the lines are minus 6 2 3 2</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">2 this is the line L what about em</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">it's 1 minus 1 2 1 2 minus 1</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">so it's projective coordinates of these</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">two lines now applying this formula so</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">there are three components the first one</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">involves the second and the third</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">entries of this three to three</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">proportions so three times minus one</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">minus two times one it's here then</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">middle entry is obtained from the first</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">and third entries so it's 2 minus 1</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">minus minus 6 times minus 1 and the</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">third entry is obtained by from the</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">first and second entry here so it's</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">minus 6 times 1 minus 3 times minus 1</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">and what we get is minus 3 minus 2 is</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">minus 5 minus 2 minus 6 is minus 8 minus</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">6 plus 3 is minus 3 then we're allowed</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">to multiply say by minus 1 so we get 5 2</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">8 2 3 now if we want to convert the that</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">point from projective coordinates to</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">affine coordinates the best thing is to</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">do is divide by this first entry so we</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">get 1 here when we divide by 5</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">everything we get 1 2 8 over 5 2 3 or 5</div>

</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">coordinates are so eight fifths and</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">three ifs it is here and 3/5 yep</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">so notice that with this projective</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">point of view these two problems of</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">finding the line that goes through two</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">points and the point line two lines</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">these are essentially exactly the same</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">problems we get kind of duality</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">similarity between points and lines that</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">was completely invisible to us from a</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">fine point of view so that's a powerful</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">kind of orientation and will be useful</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">for us so we've learnt a lot</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">this lecture reminder that this lecture</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">is complemented by a lot of extra</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">information on the course there are</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">material given in details and justified</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">some theorems formulas that we talked</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">about a lot of work examples and</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">problems for you to tackle with a range</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">of difficulties and some even more</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">advanced problems occasionally so there</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">are a lot to do and it's a great subject</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">so I hope you enjoyed this lecture in</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">algebraic calculus I'm Adam Scott thanks</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">for listening</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">[Music]</div>

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<div id="label-text" class="style-scope yt-dropdown-menu">English (auto-generated)</div>

</div>

</div>

Projective Coordinates for Points and Lines

Attachments