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Current time:0:00Total duration:2:21

Video transcript

now that we've seen how Bezier curves behave geometrically let's take a look at the algebra starting with a three-point polygon as before we construct a point Q using linear interpolation that is a weighted average on the line segment a B algebraically Q can be written as Q equals 1 minus T times a plus T times B next we construct a point on the line segment BC which means that R can be written as R equals 1 minus T times B plus T times C finally we connect Q R and do one final linear interpolation to get P our point on the curve P equals 1 minus T times Q plus T times R from this last equation it kind of looks like P is degree 1 in T but the first two equations also depend on T so let's substitute the first two equations into the third to get this combined expression multiplying out the terms and collecting I can rewrite P as P equals 1 minus T squared times a plus 2t times 1 minus T times B plus T squared times C all those squared terms show us that P is actually a degree 2 polynomial interesting a three-point polygon leads to a degree 2 polynomial that kind of makes sense because we did two stages of linear interpolation and the first stage we computed qnr and in the second stage we computed P now what happens to the degree if we start with a four point polygon can you guess in the first stage I compute three points using linear interpolation in the second stage I compute two points and in the third stage I compute one point since I have three stages the resulting curve will be degree three that means a four point polygon results in a degree three curve you can generalize to Castle shows algorithm to start with five six or any number of points the rule is if we start with n points you get a polynomial of degree n minus one pretty neat and congratulations on finishing this lesson if you're feeling particularly bold try your hand at the following bonus challenge