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Catmull-Rom
splines interpolate a set of **control
points**, and
are guaranteed to* pass* through the
control points (except for the first and last points).

A Catmull-Rom spline has the following properties:

Since there are four constraints, we will have four unknowns in the spline equation (i.e., we will have a cubic spline ).

Cubic spline equation:

The goal is to derive**a**, **b**, **c** and **d** using the constraints. This will
then give the precise form of **x(t)**.

Four linear equations with four unknowns:

This is easily solved using algebra to determine**a**, **b**, **c**, and **d**!

The four equations can be solved, giving the result:

Given four control points (**p0, p1, p2, p3**), we
can compute **a**,** b**, **c,** and
**d**. Then, we can sample the curve

at regular intervals along the interval t=[0, 1] to generate points along the spline spanning the space between** p1** and **p2**.

A Catmull-Rom spline has the following properties:

- It passes through the middle two points, and interpolates the space between them

- The tangents at p1 and p2 are:

Since there are four constraints, we will have four unknowns in the spline equation (i.e., we will have a cubic spline ).

Cubic spline equation:

The goal is to derive

Four linear equations with four unknowns:

This is easily solved using algebra to determine

The four equations can be solved, giving the result:

Given four control points (

at regular intervals along the interval t=[0, 1] to generate points along the spline spanning the space between

High

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