Curves and Surfaces

Design software stores a curved letter, a car panel, or a font glyph not as a dense list of points but as a handful of control points. The curve is recovered from them by a fixed rule, and moving one control point reshapes only a local piece. Bézier curves are the rule that made this practical, and they are built entirely from affine and convex combinations of the control points.¹

Bézier curves

A Bézier curve is a combination of control points with polynomials in $t$ as weights, for $0\le t\le 1$. A quadratic curve uses three control points; a cubic uses four:

At each $t$ the polynomial weights are nonnegative and sum to $1$ — for the cubic, $(1-t{)}^3+3t(1-t{)}^2+3t^2(1-t)+t^3=1$ by the binomial theorem. Every point of the curve is a convex combination of the control points, so the curve lies inside their convex hull. It passes through the first and last control points, at $t=0$ and $t=1$, but generally not through the interior ones; those pull the curve toward themselves without lying on it.

The convex-hull property is inherited by any affine image of the curve. If $A$ is a matrix, then $A\mathbf{x}(t)$ is the Bézier curve on the transformed control points $A{\mathbf{p}}_0,\dots ,A{\mathbf{p}}_3$, because matrix multiplication distributes over the combination. Rotating, scaling, or shearing a curve amounts to transforming four points and redrawing.

Tangents at the endpoints

The control polygon governs the direction of the curve at its ends. For a parametric curve, the tangent direction at $\mathbf{w}(t)$ is the derivative ${\mathbf{w}}^{\prime }(t)$, taken coordinate by coordinate.

The de Casteljau algorithm

A Bézier curve can be evaluated without expanding any polynomials, using only repeated linear interpolation between control points. At parameter $t$, replace each consecutive pair of points by the point a fraction $t$ of the way between them; repeat on the shorter list until one point remains, which is $\mathbf{x}(t)$.

At $t=\frac{1}{2}$ every interpolation is a midpoint, which makes the construction easy to follow. For a cubic with control points ${\mathbf{p}}_0,{\mathbf{p}}_1,{\mathbf{p}}_2,{\mathbf{p}}_3$, the first round of midpoints is ${\mathbf{q}}_i=\frac{1}{2}({\mathbf{p}}_i+{\mathbf{p}}_{i+1})$; the second is ${\mathbf{r}}_i=\frac{1}{2}({\mathbf{q}}_i+{\mathbf{q}}_{i+1})$; the third is the curve point $\mathbf{x}(\frac{1}{2})=\frac{1}{2}({\mathbf{r}}_0+{\mathbf{r}}_1)$.

The construction also splits the curve. The points ${\mathbf{p}}_0,{\mathbf{q}}_0,{\mathbf{r}}_0,\mathbf{x}(\frac{1}{2})$ are the control points of the piece from $t=0$ to $t=\frac{1}{2}$, and $\mathbf{x}(\frac{1}{2}),{\mathbf{r}}_1,{\mathbf{q}}_2,{\mathbf{p}}_3$ control the piece beyond. Repeated subdivision produces control polygons that converge to the curve, the basis for drawing a Bézier curve as a short chain of line segments.

Joining curves

Complex outlines are built from many short Bézier segments joined end to end. The smoothness of a join is graded by how much the two segments agree at the shared point.

• $G^0$ (geometric) continuity: the segments meet, with the terminal point of the first equal to the initial point of the second. Without more, a corner may appear.
• $G^1$ continuity: the tangent vectors at the join point in the same direction, though their lengths may differ. No corner, but the parameterization may jump in speed.
• $C^1$ (parametric) continuity: the tangent vectors at the join are equal, not merely parallel. The strongest of the three, and the usual target.

The continuity conditions translate into constraints on the control points.

Matching second derivatives as well gives $C^2$ continuity, which is possible for cubics but tightly constrains the control points. B-splines relax this by sharing three control points between adjacent segments, gaining $C^2$ smoothness at the cost of no longer passing through any control point.

Matrix form

A Bézier curve is a linear combination of control points with polynomial weights, so it factors into a matrix product. For the cubic, collect the control points into a geometry matrix $G=[{\mathbf{p}}_0{\mathbf{p}}_1{\mathbf{p}}_2{\mathbf{p}}_3]$ and the powers of $t$ into $\mathbf{u}(t)=(1,t,t^2,t^3)$:

The Bézier basis matrix $M_B$ holds the coefficients of the blending polynomials $(1-t{)}^3$, $3t(1-t{)}^2$, $3t^2(1-t)$, $t^3$, expanded in powers of $t$. Multiplying $M_B\mathbf{u}(t)$ recovers the four weights; multiplying by $G$ applies them to the control points. Other cubic families reuse the same factored form with a different basis matrix: a Hermite curve replaces $M_B$ by a Hermite basis matrix and stores tangent vectors instead of interior control points, and a B-spline uses the B-spline basis matrix.

Bézier surfaces

A surface patch is the two-parameter analogue. Take a $4\times 4$ grid of control points ${\mathbf{p}}_{ij}$ and let each of the four rows control a Bézier curve in the parameter $t$; the four resulting curves are then blended by a second Bézier combination in a parameter $s$. The patch is

with $b_1,\dots ,b_4$ the four cubic blending polynomials. The sixteen control points form a web that the surface bends toward, and the surface stays inside their convex hull.

The essential structure remains the affine combination. A Bézier object is a weighted average of control points with weights that are nonnegative and sum to one, which places every curve and surface in the convex hull of its control net and returns the geometry of this module to the algebra of affine combinations.