Bezier Curves

Overview

Linear, quadratic, and cubic Bezier curve evaluation in 2D, 3D, and 4D. Includes first and second derivatives (velocity/acceleration), curve splitting via De Casteljau’s algorithm, and arc length approximation.

Why Use This?

Bezier curves are used for smooth paths, animation curves, font rendering, vector graphics, camera tracks, and UI transitions.

Functions

All functions available in _f64 and _f32 variants. Suffix 2, 3, 4 indicates dimensionality.

Curve Evaluation

Linear Bezier (2 control points)

B(t) = (1-t)*P0 + t*P1

Function	Description
`lm2_bezier_linear2_f32(p0, p1, t)`	2D linear Bezier
`lm2_bezier_linear3_f32(p0, p1, t)`	3D linear Bezier
`lm2_bezier_linear4_f32(p0, p1, t)`	4D linear Bezier

Quadratic Bezier (3 control points)

B(t) = (1-t)^2*P0 + 2*(1-t)*t*P1 + t^2*P2

Function	Description
`lm2_bezier_quadratic2_f32(p0, p1, p2, t)`	2D quadratic
`lm2_bezier_quadratic3_f32(p0, p1, p2, t)`	3D quadratic
`lm2_bezier_quadratic4_f32(p0, p1, p2, t)`	4D quadratic

Cubic Bezier (4 control points)

B(t) = (1-t)^3*P0 + 3*(1-t)^2*t*P1 + 3*(1-t)*t^2*P2 + t^3*P3

Function	Description
`lm2_bezier_cubic2_f32(p0, p1, p2, p3, t)`	2D cubic
`lm2_bezier_cubic3_f32(p0, p1, p2, p3, t)`	3D cubic
`lm2_bezier_cubic4_f32(p0, p1, p2, p3, t)`	4D cubic

Derivatives

First Derivative (Velocity)

Function	Description
`lm2_bezier_quadratic_derivative2_f32(p0, p1, p2, t)`	2D quadratic velocity
`lm2_bezier_quadratic_derivative3_f32(p0, p1, p2, t)`	3D quadratic velocity
`lm2_bezier_quadratic_derivative4_f32(p0, p1, p2, t)`	4D quadratic velocity
`lm2_bezier_cubic_derivative2_f32(p0, p1, p2, p3, t)`	2D cubic velocity
`lm2_bezier_cubic_derivative3_f32(p0, p1, p2, p3, t)`	3D cubic velocity
`lm2_bezier_cubic_derivative4_f32(p0, p1, p2, p3, t)`	4D cubic velocity

Second Derivative (Acceleration)

Function	Description
`lm2_bezier_cubic_second_derivative2_f32(p0, p1, p2, p3, t)`	2D cubic acceleration
`lm2_bezier_cubic_second_derivative3_f32(p0, p1, p2, p3, t)`	3D cubic acceleration
`lm2_bezier_cubic_second_derivative4_f32(p0, p1, p2, p3, t)`	4D cubic acceleration

Splitting

Split a cubic Bezier at parameter t into two cubic curves using De Casteljau’s algorithm.

Signature:

void lm2_bezier_cubic_split2_f32(
    lm2_v2_f32 p0, lm2_v2_f32 p1, lm2_v2_f32 p2, lm2_v2_f32 p3,
    float t,
    lm2_v2_f32* left_p0, lm2_v2_f32* left_p1, lm2_v2_f32* left_p2, lm2_v2_f32* left_p3,
    lm2_v2_f32* right_p0, lm2_v2_f32* right_p1, lm2_v2_f32* right_p2, lm2_v2_f32* right_p3
);

Also available in 3D (_split3_) and 4D (_split4_).

Arc Length

Approximate arc length of a cubic Bezier curve using segment subdivision.

Function	Description
`lm2_bezier_cubic_length2_f32(p0, p1, p2, p3, segments)`	2D cubic arc length
`lm2_bezier_cubic_length3_f32(p0, p1, p2, p3, segments)`	3D cubic arc length
`lm2_bezier_cubic_length4_f32(p0, p1, p2, p3, segments)`	4D cubic arc length

Parameters:

segments — Number of segments for approximation (higher = more accurate)

All arc length functions also available in _f64 variants.

Example

lm2_v2_f32 p0 = lm2_v2_make_f32(0, 0);
lm2_v2_f32 p1 = lm2_v2_make_f32(0, 10);
lm2_v2_f32 p2 = lm2_v2_make_f32(10, 10);
lm2_v2_f32 p3 = lm2_v2_make_f32(10, 0);

// Evaluate curve at midpoint
lm2_v2_f32 mid = lm2_bezier_cubic2_f32(p0, p1, p2, p3, 0.5f);

// Get velocity at midpoint
lm2_v2_f32 vel = lm2_bezier_cubic_derivative2_f32(p0, p1, p2, p3, 0.5f);

// Approximate arc length
float length = lm2_bezier_cubic_length2_f32(p0, p1, p2, p3, 100);