---
title: "Bezier surface equation and properties"
subject: "Computer Aided Design"
chapter: 4
key: "semester_4/computer_aided_design:bezier-surface-properties"
source: https://www.bctengineer.com/q/semester_4/computer_aided_design/bezier-surface-properties
site: BCT Engineering
---

# Bezier surface equation and properties

Computer Aided Design · Chapter 4: Representation of Surfaces and Solids

- Asked in: year not given, 6 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/computer_aided_design/bezier-surface-properties
- Whole chapter: https://www.bctengineer.com/md/semester_4/computer_aided_design/chapter-4

## Question

Explain the Bezier surface. Write the equation of a bicubic Bezier surface patch and list its properties. How does it differ from a Bezier curve?

## Short answer

A bicubic Bezier surface is defined by a 4x4 net of control points via P(u,w)=sum sum B_ij J_3,i(u)J_3,j(w). Properties: corner points interpolate, boundary curves are Bezier curves, convex hull, global control, affine invariance. It differs from a curve by having two parameters and a net of points rather than a polygon.

## Answer

A **Bezier surface** is a parametric surface patch defined by a rectangular net (mesh) of control points. It is the tensor product of two Bezier curves, one in the $u$ direction and one in the $w$ direction.

#### Equation
For an $(m+1)\times(n+1)$ net of control points $\mathbf B_{ij}$:

$$
\mathbf P(u,w)=\sum_{i=0}^{m}\sum_{j=0}^{n}\mathbf B_{ij}\,J_{m,i}(u)\,J_{n,j}(w),\qquad 0\le u,w\le1
$$

where $J_{m,i}(u)=\binom mi u^i(1-u)^{m-i}$ is the Bernstein basis. The **bicubic patch** has $m=n=3$, so 16 control points:

$$
\mathbf P(u,w)=\sum_{i=0}^{3}\sum_{j=0}^{3}\mathbf B_{ij}J_{3,i}(u)J_{3,j}(w)
$$

#### Properties
1. The degree in $u$ is $m$ and in $w$ is $n$ (one less than the number of points in that direction).
2. The four corner points of the net lie on the surface: $\mathbf P(0,0)=\mathbf B_{00}$, $\mathbf P(1,0)=\mathbf B_{m0}$, etc.
3. The four boundary curves are Bezier curves defined by the boundary rows and columns of the net.
4. Tangent planes at the corners are spanned by the adjacent net edges.
5. **Convex hull:** the surface lies in the convex hull of the control net.
6. Global control: moving any control point changes the whole patch.
7. Invariant under affine transformation (transform the net only).
8. Two patches join with $C^0$ continuity if the edge rows coincide, and with $C^1$ if the adjacent rows are collinear with equal ratio.

#### Difference from curve
```text
 Curve:   1 parameter u,   row of points B_i
 Surface: 2 parameters u,w, net of points B_ij

   B03---B13---B23---B33
    |     |     |     |      u -> (to the right)
   B02---B12---B22---B32
    |     |     |     |      w ^  (upwards)
   B01---B11---B21---B31
    |     |     |     |
   B00---B10---B20---B30
```
Only the four corner points lie on the surface; the other points only pull it towards them.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. How is a Bezier surface built?
   A. Sum of Hermite curves
   B. Rotating a Bezier curve
   C. Joining 4 lines
   D. Tensor product of two Bezier curves
   Answer: D. Uses Bernstein functions in u and w. (Key point: Tensor product)

2. How many control points in a bicubic patch?
   A. 16
   B. 9
   C. 12
   D. 4
   Answer: A. 4 x 4 net. (Key point: Control points)

3. Which net points lie on the surface?
   A. All
   B. Only the four corners
   C. None
   D. Boundary rows only
   Answer: B. Interior points only pull; boundary curves are Bezier curves but only corners are interpolated points of the net. (Key point: Corners)

4. What kind of control does it have?
   A. Local
   B. None
   C. Global
   D. Row-wise
   Answer: C. Moving any point changes the whole patch. (Key point: Control)

## Related in Computer Aided Design

- [Bezier curve: equation and properties](https://www.bctengineer.com/question.md?key=semester_4%2Fcomputer_aided_design%3Abezier-curve-properties) — Computer Aided Design, chapter 3
- [Biquadratic Bezier surface point calculation](https://www.bctengineer.com/question.md?key=semester_4%2Fcomputer_aided_design%3Abezier-surface-numerical) — Computer Aided Design, chapter 4
- [B-spline curve versus Bezier curve](https://www.bctengineer.com/question.md?key=semester_4%2Fcomputer_aided_design%3Abspline-vs-bezier) — Computer Aided Design, chapter 3
- [Coons patch definition and point calculation](https://www.bctengineer.com/question.md?key=semester_4%2Fcomputer_aided_design%3Acoons-patch-numerical) — Computer Aided Design, chapter 4
- [Surface of revolution: equation, points, area](https://www.bctengineer.com/question.md?key=semester_4%2Fcomputer_aided_design%3Asurface-of-revolution-numerical) — Computer Aided Design, chapter 4

## Related in other subjects

- [Derive cubic Bezier equation and properties](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Abezier-cubic-equation-properties) — Computer Graphics, chapter 5
- [Bezier curve and its properties](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Abezier-curve-properties) — Computer Graphics, chapter 5, asked 6×
- [Bezier equation and point at u=0.8](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Abezier-equation-point-u-08) — Computer Graphics, chapter 5
- [Bezier properties; point at u=0.2](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Abezier-properties-point-u-0-2) — Computer Graphics, chapter 5, asked 2×