This Week Week Topic Week 1 Week 2 Week 3 Week 4 Week 5 Coordinate Systems, Basic Functions Week 2 Trigonometry and Vectors (Part 1) Week 3 Vectors (Part 2) Week 4 Vectors (Part 3: Locus) Week 5 Tutorial A: Question and answer session for weeks 1-4 Week 6 Matrices (Part 1) Week 7 Matrices (Part 2) and Transformations Week 8 Complex numbers Week 9 Curves Week 10 Tutorial B: Question and answer session focusing on weeks 6-9

Short Course in: Mathematics and Analytic Geometry Week 9 Curves

LERP – Linear Interpolation Given two position vectors r1 and r2, a linear interpolation is a straight line joining their respective points P1 and P2.

LERP – Linear Interpolation Oh! But, hang on a minute! LERP is just the line equation defined in the interval: 0  t  1:

SLERP – Spherical Linear Interpolation SLERP is an extension on LERP to the case of a curve interpolation on a spheroid. In the most simplest case, we have two orthogonal unit position vectors r0 and r1 and we interpolate between 0 and π/2 radians with a parameter in the interval 0  t  1.

SLERP – Spherical Linear Interpolation In the general case, position vectors r0 and r1 are not orthogonal and we want to interpolate between 0 and some angle .

SLERP – Spherical Linear Interpolation The general SLERP Formula is derived as follows:

Quaternion SLERP Instead of vectors we can plug in unit quaternion to interpolate unit quaternion at parametric angles: Where:

Parametric Curves Any 2D curve can be projected in 3D space onto a parametric plane:

Parametric Curves Similarly, in polar coordinates where f() define radius length: Or in cylindrical space (in this case, an elliptic spiral):

Bezier Curves However, some parametric curves can be inflexible and difficult to design. Ideally, we would like to construct curves in a predictable way from some fixed points. We could join points with LERPs to form a piecewise curve:

Bezier Curves But, the curve is not smooth: Bezier curves can be constructed as a recursive system of LERPs (De Casteljau's algorithm). For example:

Bezier Curves

Bezier Curves The general form of Bezier curves follows a binomial expansion pattern:

Inverse Bezier Let us assume we want a cubic Bezier curve: That interpolates points p1 and p2 at times t1 and t2 respectively (remember, the ends are fixed): We solve for control points r1 and r2, by finding the inverse of the matrix:

Catmull-Rom Spline Using Tangents like this, curves can be joined with C1 continuity.

Cubic B-Splines With B-Splines, continuity is always one degree lower than the degree of each curve piece. Therefore, a cubic B-Spline has continuity C2. The following curve connects pi to pi+1, for 0t 1: Cubic B-Splines do not interpolate their end points.

Cubic B-Splines In general, a B-Spline can be defined as follows (the ti are called knots):

Bezier Surfaces Bezier curves can be extended to surfaces on unit squares.

B-Spline Surfaces