Pixar in a Box
- Start here!
- 1. Mathematics of linear interpolation
- Linear interpolation
- 2. Repeated linear interpolation
- 3. De Casteljau's algorithm
- Constructing curves using repeated linear interpolation
- 4. What degree are these curves?
- Bonus: Equations from de Casteljau's algorithm
Bonus! In this video we'll connect the degree of these curves to the number of control points in the construction.
Want to join the conversation?
- this is too hard for me and what is the difference between linear interpulation and a beziar curve?(14 votes)
- 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.(6 votes)
- I dont understand any of this(6 votes)
- for Q=(t-1)A+tB can A swap with B but still have the same answer?(4 votes)
- 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)
- Is the variable A, B and C the angle or the point on a coordinate plain, or something else?(3 votes)
- 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)
- I think the only people understanding this are animators and or mathematicians/people godly at math.(2 votes)
- 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.