Essential Mathematics for Computational Design

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.

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​.

So, how do we define the end points of a line segment that represents a given vector? Let us define an anchor point (A) so that:

A = (1, 2, 3)

And a vector:

v ​= <5, 6, 7>

The tip point (B) of the vector is calculated by adding the corresponding components from anchor point and vector ​v​:

B = A + ​v

B = (1+5, 2+6, 3+7)

B = (6, 8, 10)

Figure 2: The relationship between a vector, the vector anchor point, and the point coinciding with the vector tip location.

Position vector

One special vector representation uses the origin point (0,0,0) as the vector anchor point. The position vector ​v ​= <a1,a2,a3> is represented with a line segment between two points, the origin and B, so that:

Origin point = (0,0,0)

B = (a1,a2,a3)

Figure 3: Position vector. The tip point coordinates equal the corresponding vector components.

A position vector for a given vector ​v​= < a1,a2,a3 > is a special line segment representation from the origin point (0,0,0) to point (a1,a2,a3).

Vectors vs. points

Do not confuse vectors and points. They are very different concepts. Vectors, as we mentioned, represent a quantity that has direction and length, while points indicate a location. For example, the North direction is a vector, while the North Pole is a location (point). If we have a vector and a point that have the same components, such as:

v​ = <3,1,0>

P = (3,1,0)

We can draw the vector and the point as follows:

Figure 4: A vector defines a direction and length. A point defines a location.

Vector length

As mentioned before, vectors have length. We will use |​a​| to notate the length of a given vector ​a​. For example:

a ​= <4, 3, 0>

|​a​| = √(4​2​ + 3​2​ + 0​2​)

|​a​| = 5

In general, the ​length​ of a vector ​a​<a1,a2,a3>​ ​is​ ​calculated as follows:

|​a​| = √(a1​^2​ + a2​^2​ + a3​^2​)

Figure 5: Vector length.

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.