What is a Bézier patch?

Bézier patch, similar to Bézier curve, is defined by a grid of control points. The grid can be of any size. The most common form of Bézier patch is bicubic Bézier surface patch, which uses a $4 \times 4$ control point grid ($P$) to define the surface.

Note: The bigger the grid, the more complex shapes we can make.

The Bézier patch is used in 3D modeling, such as Computer-aided design and computer graphics. It finds its applications in video games and animations.

How does it work?

• The control points in the Bézier patch work as a magnet, pulling the surface of the patch towards them.

• Each Bézier curve is made of $4$ control points, and the Bézier patch is made up of $16$ control points ($4$ Bézier curves).

• The shape of the surface is calculated based on the two parameters, named $u$ and $v$.

• You can think of $u$ and $v$ as coordinates on the map, specifying where we are on the surface of the patch.

• Each control point in the grid has a weight assigned to it based on the $u$ and $v$. These weights are usually the values of the control points.

• The closer $u$ or $v$ is to the position of a control point, the more weight that point has, and the more it influences the shape of the surface.

To represent a Bézier patch, the equation used to calculate the position of a point on the surface at parameters $u$ and $v$ is given below.

Where,

• $u$ is the coordinate in the horizontal direction of size $1 \times 4$.

  • $u = [u^3 \quad u^2 \quad u \quad 1]$

• $v$ is the coordinate in the vertical direction of size $4 \times 1$.

  • $v = \begin{bmatrix} v^3 \\ v^2 \\ v \\ 1\\ \end{bmatrix}$

• $B$ is a matrix describing the blending functionsThe blending functions, also known as basis functions, are the Bernstein polynomials for Bézier patches for a parametric cubic curve.

Note: Each row represents the coefficient of the control points.

• $P$ is a $4\times 4$ matrix representing the control points of the surface patch.

• The matrix for the control points of the surface patch is given below.

• $B^T$ is the transpose of the $B$ matrix.

Once the weight for all the control points is calculated using the Bernstein polynomialA polynomial that is a linear combination of Bernstein basis polynomials., we can get the shape of the surface.

Note:

• The older points in the $P$ matrix are then replaced by the newly calculated values of the points.

• All other matrices remain the same in the equation.

• The $P$ matrix contains all the updated values of the control points which is then used to draw the surface.

de Casteljau algorithm

The de Casteljau algorithm is a technique used for evaluating points on a Bézier curve or Bézier surface patch. It is based on the de Casteljau algorithm, which recursively subdivides the control points of the curve or surface to calculate points at specific values.

For Bézier curves, the de Casteljau algorithm interpolates control points linearly and is a one-dimensional process. However, for Bézier patches, the de Casteljau algorithm extends it to a two-dimensional process, where points on the surface are interpolated in two directions: $u$ and $v$.

How it works

• The working of the de Casteljau algorithm is as follows.

• In order to draw a surface, define a Bézier curve using the four control points.

• Choose a value of $u$.

• At the value of $u$, evaluate the four control points as $P_1(u)$, $P_2(u)$, $P_3(u)$, $P_4(u)$for all the four curves respectively (as shown in the first diagram below).

• Take these four values as the control points and define another 1D Bézier curve.

• Choose the value of parameter $v$.

• It will give you the point $P$ on the surface.

• Change the values of $u$ and $v$ in order to draw every point on the surface.

Properties of Bézier patch

Bézier patch have several important properties, some of which are explained below.

• Interpolation: Bézier patch can interpolate their control points to ensure that the surface passes through each of the control point.

• Convex hull property: The Bézier patch lies within the convex hullA minimum convex set enclosing a sample of points. of its control points. It means that it stays close to the shape formed by its control points (no self-intersections or singularities).

• Affine invariance: Bézier patches are invariant under affine transformationsA linear mapping method that preserves points, straight lines, and planes., such as translation, rotation, scaling, and skewing. It means that any transformations applied to the control points are reflected in the patch.

• Local control: If we modify the position of one control point, it affects only a local portion of the Bézier patch. It allows each adjustments to specific parts of the surface without changing the entire shape.

Continue reading

• What is a Bézier curve?

• What is Chaikin's algorithm?