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4. What degree are these curves?

Bonus! In this video we'll connect the degree of these curves to the number of control points in the construction.

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  • mr pants teal style avatar for user GoldenSniperPro
    this is too hard for me and what is the difference between linear interpulation and a beziar curve?
    (15 votes)
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    • leaf green style avatar for user Benjamin Faust
      Linear interpolation means, that the change is at a constant rate. Imagine an electric garage door closing or sand in an hourglass running.

      Bezier curves allow you to make a change in the speed of the changes, accelerate and decelerate things. Since this is the way most things actually move, the beziers are quite essential to animation.
      (7 votes)
  • hopper cool style avatar for user wublablu
    I dont understand any of this
    (7 votes)
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  • duskpin ultimate style avatar for user Eunice Park
    My brain is popping. Can anyone explain this whole video?
    (7 votes)
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  • starky ultimate style avatar for user Eleanor Cooper
    I got confused within the first 30 seconds... where does Q=(1-t)A + tB come from??
    (7 votes)
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  • blobby green style avatar for user Michael
    linear equals straight
    curve equals curve
    (4 votes)
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  • starky ultimate style avatar for user Trung Hoang
    for Q=(t-1)A+tB can A swap with B but still have the same answer?
    (4 votes)
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    • mr pants purple style avatar for user Kelsey Chan
      The resulting curve would be different, assuming we keep R = (1-t)*B+t*C.

      Let's picture the value of Q when we give t values from 0 to 1.
      - If we use Q = (1-t)*A + t*B, then Q starts at point A and moves on the line segment toward point B.
      - If we use Q = (1-t)*B + t*A, then Q starts at point B and moves on the line segment toward point A.

      If you plug in the expressions for Q and R into the equation for P, you'll find that P = t(1-t)*A + (1-t)*B + t^2*C, which is different from the equation for P in the video.
      (0 votes)
  • duskpin seedling style avatar for user Rupesh Jadhav
    It is elaborated shortly. Please explain with more ease.
    (3 votes)
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  • piceratops ultimate style avatar for user Addison Ballif
    Is the variable A, B and C the angle or the point on a coordinate plain, or something else?
    (3 votes)
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    • duskpin ultimate style avatar for user XDmysticXD
      The variables A, B, and C are just the name of 3 points on a plane and the video explains how to construct a beziers curve using these three points. However, you could use any amount of points and still get a curve, and you could name each point a different name.
      (1 vote)
  • aqualine ultimate style avatar for user robert.hankin
    lots of 3d animation uses trigonometry. you know collage level stuff. ez stuff. unhinged laughter
    (2 votes)
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  • winston default style avatar for user Durst,Wyatt
    I think the only people understanding this are animators and or mathematicians/people godly at math.
    (2 votes)
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Video transcript

- Now that we've seen how Bézier 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 AB. Algebraically, Q can be written as Q = (1-t)A + tB Next we construct a point R on the line segment BC, which means that R can be written as R = (1-t)B + tC Finally we connect Q and R, and do one final linear interpolation to get P, out point on the curve. P = (1-t)Q + tR From this last equation, it kinda look 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 = (1-t)2*A + 2t(1-t)B +t2*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. Thar kinda makes sense because we did two stages of linear interpolation. In the first stage we computed Q and R 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 3. That means a four-point polygon results in a degree 3 curve. You can generalize deCastlejau's 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-1. Pretty neat. And congratulations on finishing this lesson. If you're feeling particularly bold, try your hand at the following bonus challenge.