Curves and Surfaces · LVA TL;DR

General

Why Not Only Triangles

Piece-wise linear geometry is easy.
Examples include line segments, triangles, and cells.
Complex smooth shapes need many tiny pieces.
Too few pieces create visible discontinuities.
Too many pieces are hard to model and animate manually.
A better approach is to define a few control points and compute intermediate curve or surface points by interpolation.

Interpolation Recap

Bilinear Interpolation

Interpolation computes a weighted sum of control points.
In 2D, use parameters $u$ and $v$ .
Bilinear interpolation works for patches and quads.
Four control points do not need to be coplanar.
Parameter values decide influence weights.
The weights sum to 1.

Trilinear Interpolation

Trilinear interpolation is the 3D version.
It uses parameters $x$ , $y$ , and $z$ .
It is used inside cube or hexahedron cells.
It interpolates from 8 corner values.
It is smoother than nearest-neighbor sampling.

Cell Types

Interpolation depends on the cell type.
Common cells:
- Vertex.
- Line.
- Triangle.
- Quad.
- Tetrahedron.
- Hexahedron.

Piece-Wise Linear Interpolation

Idea

Complex geometry is stitched from independent linear pieces.
Line segments approximate a curve.
Triangles approximate a surface.
Cubes or cells approximate a volume.
The result is polygonal or polyhedral.

Limitation

It is good for simple shapes.
It is bad for smooth complex surfaces unless the resolution is high.
Patch and cell borders can become visible.
Modeling and animation become inefficient.

Splines

Idea

A spline is an interpolated curve generated from control points or a control polygon.
A spline surface extends the same idea to surfaces.
Instead of manually placing every triangle:
- Place control points.
- Use basis or blending functions.
- Compute intermediate points.

Control Polygon

The control polygon connects the control points with line segments.
The curve follows the general shape of the control polygon.
It does not necessarily pass through every control point.
The beginning and end of a Bézier curve are tangent to the control polygon.

Quadratic Bézier Curves

Construction

A quadratic Bézier curve has 3 control points.
It has degree 2.
It has 3 basis functions.
It can be built by recursive midpoint subdivision.
Repeated corner cutting converges to a smooth limit curve.

Formula

p(t) = (1-t)^2p_0 + 2(1-t)t p_1 + t^2p_2

$t$ ranges from 0 to 1.
The weights are normalized and always sum to 1.
$p_0$ has maximum influence at $t=0$ .
$p_1$ has maximum influence around $t=\frac{1}{2}$ .
$p_2$ has maximum influence at $t=1$ .

Cubic Bézier Curves

Construction

A cubic Bézier curve has 4 control points.
It has degree 3.
It has 4 basis functions.
It uses the same recursive subdivision idea.
It is common in modeling because it is still cheap and flexible.

Formula

p(t) = (1-t)^3p_0 + 3(1-t)^2t p_1 + 3(1-t)t^2 p_2 + t^3p_3

Higher-Order Bézier Curves

Rule

$N$ control points produce degree $N-1$ .
The number of basis functions equals the number of control points.
Coefficients follow the binomial pattern.

Advantages

They are a simple generalization.
They are easy to implement recursively.
All weights still sum to 1.
Endpoints are controlled exactly.

Disadvantages

High-degree polynomials become inefficient and unstable.
Every control point influences almost the whole curve.
Moving one control point changes too much.
For many control points, Bézier curves are not the best choice.

Bézier Surfaces

Idea

Bézier surfaces extend Bézier curves to two parameters $u$ and $v$ .
Use a 2D grid of 3D control points.
A 2D blending function is the product of two 1D blending functions.
The surface point is a weighted sum of all control points.

Rectangular Base

Bézier surfaces work naturally for rectangular parameter domains.
Control points form a grid.
The number of control points can differ in both directions.

Trimming

Not every useful surface is rectangular.
A trimming curve restricts the parameter space.
Parameter values outside the trimming boundary are ignored.
A trimming curve can be polygonal or a spline.

B-Splines

Motivation

Bézier curves have two main problems:
- Many control points create high-degree polynomials.
- They have poor local control.
B-splines solve this with local basis functions.
Basis functions are pieces of low-degree polynomials.
Each control point influences only a local parameter region.

Uniform B-Splines

The same basis function is shifted along the parameter axis.
The parameter range is larger than 0 to 1.
Contributions of overlapping basis functions are summed.
At each parameter value, the weights still sum to 1.
Moving one control point changes only a nearby curve region.

Uniform Cubic B-Splines

A cubic basis function is made from four polynomial segments.
Each segment is evaluated in a local range from 0 to 1.
One control point influences roughly $t=i-2$ to $t=i+2$ .
The global ends are limited because not enough neighboring functions exist there.

Non-Cubic B-Splines

Cubic B-splines are only one option.
Lower-order B-splines have smaller influence regions and less smoothness.
Higher-order B-splines have larger influence regions and more smoothness.
Basis functions can be computed recursively after mapping $t$ to the local range.

NURBS

Meaning

NURBS means non-uniform rational B-splines.
Non-uniform means the knot or parameter gaps do not need to be equal.
Rational means the blending function is a ratio of two polynomials.

Advantage

NURBS can represent conics exactly.
Examples include circles, cylinders, and spheres.
They are preferred in CAD and precise modeling.
They are more practical than plain uniform B-splines for many engineering shapes.

Remember

Linear interpolation is cheap but needs many pieces.
Bézier curves are simple and smooth but have poor local control for many points.
B-splines give local control with low-degree polynomial pieces.
NURBS extend B-splines for exact conics and practical modeling.