Essential Mathematics for Computational Design
Introduces to design professionals the foundation mathematical concepts for effective development of computational 3D models.

Essential Mathematics for Computational Design introduces to design professionals the foundation mathematical concepts that are necessary for effective development of computational methods for 3D modeling and computer graphics. This is not meant to be a complete and comprehensive resource, but rather an overview of the basic and most commonly used concepts. The material is directed towards designers who have little or no background in mathematics beyond high school. All concepts are explained visually using Grasshopper® (GH), the generative modeling environment for Rhinoceros® (Rhino).

The content is divided into three chapters. Chapter 1 discusses vector math including vector representation, vector operation, and line and plane equations. Chapter 2 reviews matrix operations and transformations. Chapter 3 includes an in-depth review of parametric curves with special focus on NURBS curves and the concepts of continuity and curvature. It also reviews NURBS surfaces and polysurfaces.

I would like to acknowledge the excellent and thorough technical review by Dr. Dale Lear of Robert McNeel & Associates. His valuable comments were instrumental in producing this edition. I would also like to acknowledge Ms. Margaret Becker of Robert McNeel & Associates for reviewing the technical writing and formatting.

Rajaa Issa

Robert McNeel & Associates

1. Vector Mathematics

1.1 Vector representation
Position vector
Vectors vs. points
Vector length
Unit vector

1.2 Vector operations
Vector scalar operation
Vector subtraction
Vector properties
Vector dot product
Vector dot product, lengths, and angles
Dot product properties
Vector cross product
Cross product and angle between vectors
Cross product properties

1.5 Tutorials
Face direction
Exploded box
Tangent spheres

2. Matrices and Transformations

2.1 Matrix operations
Matrix multiplication
Identity matrix

2.2 Transformation operations
Translation (move) transformation
Rotation transformation
Scale transformation
Shear transformation
Mirror or reflection transformation
Planar Projection transformation

3. Parametric Curves and Surfaces

3.1 Parametric curve
Curve parameter
Curve domain or interval
Curve evaluation
Tangent vector to a curve
Cubic polynomial curves
Evaluating cubic Bézier curves

3.2 NURBS curves
Degree
Control points
Weights of control points
Knots
Knots are parameter values
Evaluation rule
Characteristics of NURBS curves
Evaluating NURBS curves

3.5 Parametric surfaces
Surface parameters
Surface domain
Surface evaluation
Tangent plane of a surface

3.7 Surface curvature
Principal curvatures
Gaussian curvature
Mean curvature

3.8 NURBS surfaces
Characteristics of NURBS surfaces
Singularity in NURBS surfaces
Trimmed NURBS surfaces

3.9 Polysurfaces

3.10 Tutorials
Continuity between curves
Surfaces with singularity