Curves and Surfaces

• Approximates complex geometry using simple, linear segments or patches (e.g., triangles, cubes).

• Each piece is interpolated independently.

• Limitation: requires many small patches to avoid discontinuities, making it inefficient for smooth complex surfaces.

• Motivation: improve surface modeling beyond linear interpolation by using curves.

• Splines are interpolated curves (and spline surfaces for 3D) used to model complex shapes.

• Instead of manually defining every triangle, designers specify control points or polygons.

• The intermediate geometry is generated through interpolation.

• This applies to both curves and surfaces, enabling smooth and efficient modeling and animation.

• Constructed from three control points

• Curve is created by recursive subdivision (cutting away corners)

• Each midpoint lies on the curve and is never moved

• The result is a smooth curve (limited curve)

• Formula:

• Used for interpolating geometry and modeling smooth paths

So let curve of degree n = 3 dann controll points 4

Or controll points -1 -> curve degree

Degree = number of control points – 1

• All control points affect the full curve (except endpoints)

• Control polygon = set of line segments between control points

• Start and end of the curve are tangent to the control polygon

• 3D control points yield 3D space curves

• Subdivision and recursive evaluation allow efficient implementation

• Blending functions can be computed recursively

• Bézier curves are a special type of spline (Bézier splines)

• Concept can be extended from curves to surface modeling using similar principles

Except for the initial midpoints, moving one controll point influences the whole curve.

We just calculate the blending in n and m direction and then multiply the results to get the blending

• Trimming restricts the parameter space (u, v) to a subregion using a polygon or spline.

• This allows modeling non-rectangular surface bases.

• Parameters outside the trimming polygon do not influence the surface.

• Problem with Bézier curves:

  • Many control points → high-degree polynomials → computationally expensive.

  • Every control point affects the whole curve → inefficient for local modifications.

• B-spline solution:

  • Uses low-degree piecewise polynomials as blending functions.

  • Blending functions are localized (each influences only a small region).

  • These are called basis functions and are shifted along parameter space.

  • Curve is formed by summing all control points weighted by their shifted basis function:

• NURBS = Nonuniform Rational B-Splines

• Their blending function is a ratio of two polynomials → a rational function

• Advantages:

  • Can represent conics exactly (ideal for modeling cylinders/spheres)

• Nonuniform: Gaps in parameter t can be non-uniform and non-integer (user-defined)