Essential Mathematics for Computational Design

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Essential Mathematics for Computational Design, by Rajaa Issa (Robert McNeel & Associates), introduces to design professionals the foundation mathematical concepts that are necessary for effective development of computational methods for 3-D 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.

The author 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 the document.


Vector Mathematics

Vector representation

Position vector Vectors vs. points Vector length Unit vector

Vector operations

Vector scalar operation Vector addition 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

Vector equation of line

Vector equation of a plane

Tutorials

Face direction

Input Parameters Solution

Exploded box

Input Parameters Solution

Tangent spheres

Input Parameters Solution

Matrices and Transformations

Matrix operations

Matrix multiplication

Method 1 Method 2 Identity matrix

Transformation operations

Translation (move) transformation

Rotation transformation

Scale transformation

Shear transformation

Mirror or reflection transformation

Planar Projection transformation

Tutorial

Multiple transformations

Input

Additional input

Solution

Parametric Curves and Surfaces

Parametric curves

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

NURBS curves

Degree Control points Weights of control points Knots Knots are parameter values Knot multiplicity Fully-multiple knots Uniform knots Non uniform knots Evaluation rule Characteristics of NURBS curves Clamped vs. periodic NURBS curves Weights Evaluating NURBS curves Solution

Curve geometric continuity

Curve curvature

Parametric surfaces

Surface parameters

Surface domain

Surface evaluation

Tangent plane of a surface

Surface geometric continuity

Surface curvature

Principal curvatures

Gaussian curvature

Mean curvature

NURBS surfaces

Characteristics of NURBS surfaces Singularity in NURBS surfaces Trimmed NURBS surfaces

Polysurfaces

Tutorials

Continuity between curves

Input

Parameters

Solution

Surfaces with singularity

Input

Parameters

Solution

References

Edward Angel, "Interactive Computer Graphics with OpenGL,” Addison Wesley Longman, Inc., 2000.

James D Foley, Steven K Feiner, John F Hughes, "Introduction to Computer Graphics" Addison-Wesley Publishing Company, Inc., 1997.

James Stewart, "Calculus," Wadsworth, Inc., 1991.

Kenneth Hoffman, Ray Kunze, “Linear Algebra”, Prentice-Hall, Inc., 1971

Rhinoceros® help document, Robert McNeel and Associates, 2009.