# Curves and Surfaces ## 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 ```math 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. - p0 has maximum influence at t equals 0. - p1 has maximum influence around t equals one half. - p2 has maximum influence at t equals 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 ```math 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 minus 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 equals i minus 2 to t equals i plus 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.