Revision as of 02:17, 16 December 2021

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

A vector indicates a quantity, such as velocity or force, that has direction and length. Vectors in 3D coordinate systems are represented with an ordered set of three real numbers and look like:

v = <a1, a2, a3>

Vector representation

In this document, lower case bold letters will notate vectors. Vector components are also enclosed in angle brackets. Upper case letters will notate points. Point coordinates will always be enclosed by parentheses. Using a coordinate system and any set of anchor points in that system, we can represent or visualize these vectors using a line-segment representation. An arrowhead shows the vector direction. For example, if we have a vector that has a direction parallel to the x-axis of a given 3D coordinate system and a length of 5 units, we can write the vector as follows:

v ​= <5, 0, 0>

To represent that vector, we need an anchor point in the coordinate system. For example, all of the arrows in the following figure are equal representations of the same vector despite the fact that they are anchored at different locations.

Figure 1: Vector representation in the 3-D coordinate system.

center|frame|Given a 3D vector ​v​ = < a1, a2, a3 >, all vector components a1, a2, a3 are real numbers. Also all line segments from a point A(x,y,z) to point B(x+a1, y+a2, z+a3) are equivalent representations of vector ​v​.

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.