Revision as of 18:33, 18 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.

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| = √(4² + 3² + 0²)

|a| = 5

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

|a| = √(a1² + a2² + a3²)

Figure 5: Vector length.

Unit vector

A unit vector is a vector with a length equal to one unit. Unit vectors are commonly used to compare the directions of vectors.

A unit vector is a vector whose length is equal to one unit.

To calculate a unit vector, we need to find the length of the given vector, and then divide the vector components by the length. For example:

|a = <4, 3, 0>

|a| = √(4² + 3² + 0²)

|a| = 5 unit length

If b = unit vector of a, then:

b = <4/5, 3/5, 0/5>

b = <0.8, 0.6, 0>

|b| = √(0.8² + 0.6² + 0²)

|b| = √(0.64 + 0.36 + 0)

|b| = √(1) = 1 unit length

In general:

a = <a1, a2, a3>

The unit vector of a = <a1/|a|, a2/|a|, a3/|a|>

Figure 6: Unit vector equals one-unit length of the vector.

Vector operations

Vector scalar operation

Vector scalar operation involves multiplying a vector by a number. For example:

a = <4, 3, 0>

2*a = <2*4, 2*3, 2*0>

2*a = <8, 6, 0>

Figure 7: Vector scalar operation.

In general, given vector a = <a1, a2, a3>, and a real number t

t*a = < t*a1, t*a2, t*a3 >

Vector addition

Vector addition takes two vectors and produces a third vector. We add vectors by adding their components.

Vectors are added by adding their components.

For example, if we have two vectors:

a<1, 2, 0>

b<4, 1, 3>

a+b = <1+4, 2+1, 0+3>

a+b = <5, 3, 3>

Figure 8: Vector addition.

In general, vector addition of the two vectors a and b is calculated as follows:

a = <a1, a2, a3>

b = <b1, b2, b3>

a+b = <a1+b1, a2+b2, a3+b3>

Vector addition is useful for finding the average direction of two or more vectors. In this case, we usually use same-length vectors. Here is an example that shows the difference between using same-length vectors and different-length vectors on the resulting vector addition:

Figure 9: Adding various length vectors (left). Adding same length vectors (right) to get the average direction.

Input vectors are not likely to be same length. In order to find the average direction, you need to use the unit vector of input vectors. As mentioned before, the unit vector is a vector of that has a length equal to 1.

Vector subtraction

Vector subtraction takes two vectors and produces a third vector. We subtract two vectors by subtracting corresponding components. For example, if we have two vectors a and b and we subtract b from a, then:

a<1, 2, 0>

b<4, 1, 4>

a-b = <1-4, 2-1, 0-4>

a-b = <-3, 1, -4>

If we subtract b from a, we get a different result:

b - a = <4-1, 1-2, 4-0>

b - a = <3, -1, 4>

Note that the vector b - a has the same length as the vector a - b, but goes in the opposite direction.

Figure 10: Vector subtraction.

In general, if we have two vectors, a and b, then a - b is a vector that is calculated as follows:

a = <a1, a2, a3>

b = <b1, b2, b3>

a - b = <a1 - b1, a2 - b2, a3 - b3>

Vector subtraction is commonly used to find vectors between points. So if we need to find a vector that goes from the tip point of the position vector b to the tip point of the position vector a, then we use vector subtraction (a-b) as shown in Figure 11.

Figure 11: Use vector subtraction to find a vector between two points.

Vector properties

There are eight properties of vectors. If a, b, and c are vectors, and s and t are numbers, then:

a + b = b + a

a + 0 = a

s * (a + b) = s * a + s * b

s * t * (a) = s * (t * a)

a + (b + c) = (a + b) + c

a + (-a) = 0

(s + t) * a = s * a + t * a

1 * a = a

Vector dot product

The dot product takes two vectors and produces a number.

For example, if we have the two vectors a and b so that:

a = <1, 2, 3>

b = <5, 6, 7>

Then the dot product is the sum of multiplying the components as follows:

a · b = 1 * 5 + 2 * 6 + 3 * 7

a · b = 38

In general, given the two vectors a and b:

a = <a1, a2, a3>

b = <b1, b2, b3>

a · b = a1 * b1 + a2 * b2 + a3 * b3

We always get a positive number for the dot product between two vectors when they go in the same general direction. A negative dot product between two vectors means that the two vectors go in the opposite general direction.

Figure 12: When the two vectors go in the same direction (left), the result is a positive dot product. When the two vectors go in the opposite direction (right), the result is a negative dot product.

When calculating the dot product of two unit vectors, the result is always between 1 and +1. For example:

a = <1, 0, 0>

b = <0.6, 0.8, 0>

a · b = (1 * 0.6, 0 * 0.8, 0 * 0) = 0.6

In addition, the dot product of a vector with itself is equal to that vector’s length to the power of two. For example:

a = <0, 3, 4>

a · a = 0 * 0 + 3 * 3 + 4 * 4

a · a = 25

Calculating the square length of vector a:

| a | = √(4² + 3² + 0²)

| a | = 5

| a |² = 25

Vector dot product, lengths, and angles

There is a relationship between the dot product of two vectors and the angle between them.

The dot product of two non-zero unit vectors equals the cosine of the angle between them.

In general:

a · b = | a | * | a | * cos(ө), or

a · b / (| a | * | a |) = cos(ө)

Where:

ө is the angle included between the vectors.

If vectors a and b are unit vectors, we can simply say:

a · b = cos(ө)

And since the cosine of a 90-degree angle is equal to 0, we can say:

Vectors a and b are orthogonal if, and only if, a · b = 0.

For example, if we calculate the dot product of the two orthogonal vectors, World xaxis and yaxis, the result will equal zero.

x = <1, 0, 0>

y = <0, 1, 0>

x · y = (1 * 0) + (0 * 1) + (0 * 0)

x · y = 0

There is also a relationship between the dot product and the projection length of one vector onto another. For example:

a = <5, 2, 0>

b = <9, 0, 0>

unit(b) = <1, 0, 0>

a · unit(b) = (5 * 1) + (2 * 0) + (0 * 0)

a · unit(b) = 2 (which is equal to the projection length of a onto b)

Figure 13: The dot product equals the projection length of one vector onto a non-zero unit vector.

In general, given a vector a and a non-zero vector b, we can calculate the projection length pL of vector a onto vector b using the dot product.

pL = |a| * cos(ө)

pL = a · unit(b)

Dot product properties

If a, b, and c are vectors and s is a number, then:

a · a = | a |²

a · (b + c) = a · b + a · c

0 · a = 0

a · b = b · a

(s * a) · b = s * (a · a) = a · (s * b)

Vector cross product

The cross product takes two vectors and produces a third vector that is orthogonal to both.

Figure 14: Calculating the cross product of two vectors.

For example, if you have two vectors lying on the World xy-plane, then their cross product is a vector perpendicular to the xy-plane going either in the positive or negative World z-axis direction. For example:

a = <3, 1, 0>

b = <1, 2, 0>

a × b = < (1 * 0 – 0 * 2), (0 * 1 - 3 * 0), (3 * 2 - 1 * 1) >

a × b = <0, 0, 5>

The vector a x b is orthogonal to both a and b.

You will probably never need to calculate a cross product of two vectors by hand, but if you are curious about how it is done, continue reading; otherwise you can safely skip this section. The cross product a × b is defined using determinants. Here is a simple illustration of how to calculate a determinant using the standard basis vectors:

i = <1, 0, 0>

j = <0,1, 0>

k = <0, 0, 1>

The cross product of the two vectors a<a1, a2, a3> and b<b1, b2, b3> is calculated as follows using the above diagram:

a × b = i (a2 * b3) + j (a3 * b1) + k (a1 * b2) - k (a2 * b1) - i (a3 * b2) - j (a1 * b3)

a × b = i (a2 * b3 - a3 * b2) + j (a3 * b1 - a1 * b3) + k (a1 * b2 - a2 * b1)

a × b = <a2 * b3 – a3 * b2, a3 * b1 - a1 * b3, a1 * b2 - a2 * b1 >

Cross product and angle between vectors

There is a relationship between the angle between two vectors and the length of their cross product vector. The smaller the angle (smaller sine); the shorter the cross product vector will be. The order of operands is important in vectors cross product. For example:

a = <1, 0, 0>

b = <0, 1, 0>

a × b = <0, 0, 1>

b × a = <0, 0, -1>

Figure 15: The relationship between the sine of the angle between two vectors and the length of their cross product vector.

In Rhino's right-handed system, the direction of a × b is given by the right-hand rule (where a = index finger, b = middle finger, and a × b = thumb).

In general, for any pair of 3-D vectors a and b:

| a × b | = | a | | b | sin(ө)

Where:

ө is the angle included between the position vectors of a and b

If a and b are unit vectors, then we can simply say that the length of their cross product equals the sine of the angle between them. In other words:

| a × b | = sin(ө)

The cross product between two vectors helps us determine if two vectors are parallel. This is because the result is always a zero vector.

Vectors a and b are parallel if, and only if, a x b = 0.

Cross product properties

If a, b, and c are vectors, and s is a number, then:

a × b = -b × a

(s * a) × b = s * (a × b) = a × (s * b)

a × (b + c) = a × b + a × c

(a + b) × c = a × c + b × c

a · (b × c) = (a × b) · c

a × (b × c) = (a · c) * b – (a · b) * c

Vector equation of line

The vector line equation is used in 3D modeling applications and computer graphics.

Figure 16: Vector equation of a line.

For example, if we know the direction of a line and a point on that line, then we can find any other point on the line using vectors, as in the following:

L = line

v = <a, b, c> line direction unit vector

Q = (x0, y0, z0) line position point

P = (x, y, z) any point on the line

We know that:

a = t * v --- (2)

p = q + a --- (1)

From 1 and 2:

p = q + t * v --- (3)

However, we can write (3) as follows:

<x, y, z> = <x0, y0, z0> + <t * a, t * b, t * c>

<x, y, z> = <x0 + t * a, y0 + t * b, z0 + t * c>

Therefore:

x = x0 + t * a

y = y0 + t * b

z = z0 + t * c

Which is the same as:

P = Q + t * v

Given a point Q and a direction v on a line, any point P on that line can be calculated using the vector equation of a line P = Q + t * v where t is a number.

Another common example is to find the midpoint between two points. The following shows how to find the midpoint using the vector equation of a line:

q is the position vector for point Q

p is the position vector for point P

a is the vector going from Q to P

From vector subtraction, we know that:

a = p - q

From the line equation, we know that:

M = Q + t * a

And since we need to find midpoint, then:

t = 0.5

Hence we can say:

M = Q + 0.5 * a

Figure 17: Find the midpoint between two input points.

In general, you can find any point between Q and P by changing the t value between 0 and 1 using the general equation:

M = Q + t * (P - Q)

Given two points Q and P, any point M between the two points is calculated using the equation M = Q + t * (P - Q) where t is a number between 0 and 1.

Vector equation of a plane

One way to define a plane is when you have a point and a vector that is perpendicular to the plane. That vector is usually referred to as normal to the plane. The normal points in the direction above the plane.

One example of how to calculate a plane normal is when we know three non-linear points on the plane.

In Figure 18, given:

A = the first point on the plane

B = the second point on the plane

C = the third point on the plane

And:

a = a position vector of point A

b = a position vector of point B

c = a position vector of point C

We can find the normal vector n as follows:

n = (b - a) × (c - a)

Figure 18: Vectors and planes.

We can also derive the scalar equation of the plane using the vector dot product:

n · (b - a) = 0

If:

n = <a, b, c>

b = <x, y, z>

a = <x0, y0, z0>

Then we can expand the above:

<a, b, c> · <x-x0, y-y0, z-z0 > = 0

Solving the dot product gives the general scalar equation of a plane:

a * (x - x0) + b * (y - y0) + c * (z - z0) = 0

Tutorials

All the concepts we reviewed in this chapter have a direct application to solving common geometry problems encountered when modeling. The following are step by step tutorials that use the concepts learned in this chapter using Rhinoceros and Grasshopper (GH).

Face direction

Given a point and a surface, how can we determine whether the point is facing the front or back side of that surface?

Input
1: A surface 2: A point
Parameters
The face direction is defined by the surface normal direction. We will need the following information: ● The surface normal direction at a surface location closest to the input point. ● A vector direction from the closest point to the input point. Compare the above two directions, if going the same direction, the point is facing the front side, otherwise it is facing the back.
Solution
1: Find the closest point location on the surface relative to the input point using the Pull component. This will give us the uv location of the closest point, which we can then use to evaluate the surface and find its normal direction.
2: We can now use the closest point to draw a vector going towards the input point. We can also draw:
3: We can compare the two vectors using the dot product. If the result is positive, the point is in front of the surface. If the result is negative, the point is behind the surface.

The above steps can also be solved using other scripting languages.

Using the Grasshopper VB component

Using the Grasshopper C# component

Using the Grasshopper Python component and the RhinoCommon SDK

Using the Grasshopper Python component and the RhinoScriptSyntax Library

Exploded box

The following tutorial shows how to explode a polysurface. This is what the final exploded box looks like:

Input
Identify the input, which is a box. We will use the Box parameter in GH.
Parameters
Think of all the parameters we need to know in order to solve this tutorial. ● The center of explosion. ● The box faces we are exploding. ● The direction in which each face is moving. Once we have identified the parameters, it is a matter of putting it together in a solution by piecing together the logical steps to reach an answer.
Solution
1: Find the center of the box using the Box Properties component in GH:
2: Extract the box faces with the Deconstruct Brep component:
3: The direction we move the faces is the tricky part. We need to first find the center of each face, and then define the direction from the center of the box towards the center of each face as follows:
4: Once we have all the parameters scripted, we can use the Move component to move the faces in the appropriate direction. Just make sure to set the vectors to the desired amplitude, and you will be good to go.

The above steps can also be solved using VB script, C# or Python. Following is the solution using these scripting languages.

Using the Grasshopper VB component

Using the Grasshopper C# component

Using the Grasshopper Python component

Tangent spheres

This tutorial will show how to create two tangent spheres between two input points. This is what the result looks like:

Input
Two points (A and B) in the 3-D coordinate system.
Parameters
The following is a diagram of the parameters that we will need in order to solve the problem: ● A tangent point D between the two spheres, at some t parameter (0-1) between points A and B. ● The center of the first sphere or the midpoint C1 between A and D. ● The center of the second sphere or the midpoint C2 between D and B. ● The radius of the first sphere (r1) or the distance between A and C1. ● The radius of the second sphere (r2) or the distance between D and C2.
Solution
1: Use the Expression component to define point D between A and B at some parameter t. The expression we will use is based on the vector equation of a line: D = A + t(B-A). ● B-A: is the vector that goes from B to A using the vector subtraction operation. ● t(B-A): where t is between 0 and 1 to get us a location on the vector. ● A+t*(B-A): gets a point on the vector between A and B.
2: Use the Expression component to also define the mid points C1 and C2.
3: The first sphere radius (r1) and the second sphere radius (r2) can be calculated using the Distance component.
4: The final step involves creating the sphere from a base plane and radius. We need to make sure the origins are hooked to C1 and C2 and the radius from r1 and r2.

The above steps can also be solved as follows.

Using the Grasshopper VB component

Using the Grasshopper C# component

Using the Grasshopper Python component

Matrices and Transformations

Transformations refer to operations such as moving (also called translating), rotating, and scaling objects. They are stored in 3D programming using matrices, which are nothing but rectangular arrays of numbers. Multiple transformations can be performed very quickly using matrices. It turns out that a [4x4] matrix can represent all transformations. Having a unified matrix dimension for all transformations saves calculation time.

Matrix operations

The one operation that is most relevant in computer graphics is matrix multiplication. We will explain it with some detail.

Matrix multiplication

Matrix multiplication is used to apply transformations to geometry. For example if we have a point and would like to rotate it around some axis, we use a rotation matrix and multiply it by the point to get the new rotated location.

Most of the time, we need to perform multiple transformations on the same geometry. For example, if we need to move and rotate a thousand points, we can use either of the following methods.

Method 1
1. Multiply the move matrix by 1000 points to move the points. 2. Multiply the rotate matrix by the resulting 1000 points to rotate the moved points. Number of operations = 2000.
Method 2
1. Multiply the rotate and move matrices to create a combined transformation matrix. 2. Multiply the combined matrix by 1000 points to move and rotate in one step. Number of operations = 1001.

Notice that method 1 takes almost twice the number of operations to achieve the same result. While method 2 is very efficient, it is only possible if both the move and rotate matrices are [4x4]. This is why in computer graphics a [4x4] matrix is used to represent all transformations, and a [4x1] matrix is used to represent points.

Three-dimensional modeling applications provide tools to apply transformations and multiply matrices, but if you are curious about how to mathematically multiply matrices, we will explain a simple example. In order to multiply two matrices, they have to have matching dimensions. That means the number of columns in the first matrix must equal the number of rows of the second matrix. The resulting matrix has a size equal to the number of rows from the first matrix and the number of columns from the second matrix. For example, if we have two matrices, M and P, with dimensions equal to [4x4] and [4x1] respectively, then there resulting multiplication matrix M · P has a dimension equal to [4x1] as shown in the following illustration:

Identity matrix

The identity matrix is a special matrix where all diagonal components equal 1 and the rest equal 0.

The main property of the identity matrix is that if it is multiplied by any other matrix, the values multiplied by zero do not change.

Transformation operations

Most transformations preserve the parallel relationship among the parts of the geometry. For example collinear points remain collinear after the transformation. Also points on one plane stay coplanar after transformation. This type of transformation is called an affine transformation.

Translation (move) transformation

Moving a point from a starting position by certain a vector can be calculated as follows:

P' = P + V

Suppose:

P(x,y,z) is a given point

v<a,b,c> is a translation vector

Then:

P'(x) = x + a

P'(y) = y + b

P'(z) = z + c

Points are represented in a matrix format using a [4x1] matrix with a 1 inserted in the last row. For example the point P(x,y,z) is represented as follows:

Using a [4x4] matrix for transformations (what is called a homogenous coordinate system), instead of a [3x3] matrices, allows representing all transformations including translation. The general format for a translation matrix is:

For example, to move point P(2,3,1) by vector v<2,2,2>, the new point location is:

P’ = P + v = (2+2, 3+2, 1+2) = (4, 5, 3)

If we use the matrix form and multiply the translation matrix by the input point, we get the new point location as in the following:

Similarly, any geometry is translated by multiplying its construction points by the translation matrix. For example, if we have a box that is defined by eight corner points, and we want to move it 4 units in the x-direction, 5 units in the y-direction and 3 units in the z- direction, we must multiply each of the eight box corner points by the following translation matrix to get the new box.

Figure 19: Translate all box corner points.

Rotation transformation

This section shows how to calculate rotation around the z-axis and the origin point using trigonometry, and then to deduce the general matrix format for the rotation.

Take a point on x,y plane P(x,y) and rotate it by angle(b).

From the figure, we can say the following:

x = d cos(a) ---(1)

y = d sin(a) ---(2)

x' = d cos(b+a) ---(3)

y' = d sin(b+a) --- (4)

Expanding x' and y' using trigonometric identities for the sine and cosine of the sum of angles:

x' = d cos(a)cos(b) - d sin(a)sin(b) ---(5)

y' = d cos(a)sin(b) + d sin(a)cos(b) ---(6)

Using Eq 1 and 2:

x' = x cos(b) - y sin(b)

y' = x sin(b) + y cos(b)

The rotation matrix around the z-axis looks like:
The rotation matrix around the x-axis by angle b looks like:
The rotation matrix around the y-axis by angle b looks like:

For example, if we have a box and would like to rotate it 30 degrees, we need the following:

1. Construct the 30-degree rotation matrix. Using the generic form and the cos and sin values of 30-degree angle, the rotation matrix will look like the following:
2. Multiply the rotation matrix by the input geometry, or in the case of a box, multiply by each of the corner points to find the box's new location.

Figure 20: Rotate geometry.

Scale transformation

In order to scale geometry, we need a scale factor and a center of scale. The scale factor can be uniform scaling equally in x-, y-, and z-directions, or can be unique for each dimension.

Scaling a point can use the following equation:

P' = ScaleFactor(S) * P

Or:

P'.x = S_x * P.x

P'.y = S_y * P.y

P'.z = S_z * P.z

This is the matrix format for scale transformation, assuming that the center of scale is the World origin point (0,0,0):

For example, if we would like to scale a box by 0.25 relative to the World origin, the scale matrix will look like the following:

Figure 21: Scale geometry.

Shear transformation

Shear in 3D is measured along a pair of axes relative to a third axis. For example, a shear along a zaxis will not change geometry along that axis, but will alter it along x and y. Here are few examples of shear matrices:

1. Shear in x and z, keeping the y-coordinate fixed:

2. Shear in y and z, keeping the x-coordinate fixed:

3. Shear in x and y, keeping the z-coordinate fixed:

Figure 22: Shear Matrices.

Mirror or reflection transformation

The mirror transformation creates a reflection of an object across a line or a plane. 2-D objects are mirrored across a line, while 3-D objects are mirrored across a plane. Keep in mind that the mirror transformation flips the normal direction of the geometry faces.

Planar Projection transformation

Intuitively, the projection point of a given 3-D point P(x,y,z) on the world xy-plane equals P_xy(x,y,0) setting the z value to zero. Similarly, a projection to xz-plane of point P is P_xz(x,0,z). When projecting to yz-plane, P_xz = (0,y,z). Those are called [[Bézier curve|orthogonal projections]].

If we have a curve as an input, and we apply the planar projection transformation, we get a curve projected to that plane. The following shows an example of a curve projected to xyplane with the matrix format.

Note: NURBS curves (explained in the next chapter) use control points to define curves. Projecting a curve amounts to projecting its control points.

Figure 24: Projection matrices.

Tutorial

Multiple transformations

Use one matrix to transform geometry as follows:

First, move input geometry so that it's center is at the origin, then rotate 45 degrees around the z axis, then scale uniformly by 0.2, then move back to original location.

Performance note: With big number of points or objects to transform, it is much more efficient to create one master transformation matrix (multiply all matrices first), then use resulting master matrix once to transform all input, as opposed to transforming multiple times with one matrix at a time.

Input
1: Objects to transform 2: Rotation angle (45 degrees) and scale factor (0.2).
Additional Input
Initial move needs the following: ● Vector from the bounding box center of input to origin.
Solution
1. Generate all transform matrices: move, rotate, scale and Inverse of initial move. 2. Multiply the matrices from last to first to generate master transformation matrix 3. Transform input using the master transformation matrix.

The above steps can also be solved using scripting.

Using the Grasshopper VB component

Using the Grasshopper C# component

Using the Grasshopper Python component and the RhinoCommon SDK

Parametric Curves and Surfaces

Suppose you travel every weekday from your house to your work. You leave at 8:00 in the morning and arrive at 9:00. At each point in time between 8:00 and 9:00, you would be at some location along the way. If you plot your location every minute during your trip, you can define the path between home and work by connecting the 60 points you plotted. Assuming you travel the exact same speed every day, at 8:00 you would be at home (start location), at 9:00 you would be at work (end location) and at 8:40 you would at the exact same location on the path as the 40th plot point.

Congratulations, you have just defined your first parametric curve! You have used time as a parameter to define your path, and hence you can call your path curve a parametric curve. The time interval you spend from start to end (8 to 9) is called the curve domain or interval.

In general, we can describe the x, y, and z location of a parametric curve in terms of some parameter t as follows: x = x(t) y = y(t) z = z(t) Where: t is a range of real numbers.

We saw earlier that the parametric equation of a line in terms of parameter t is defined as: x = x’ + t * a y = y’ + t * b z = z’ + t * c Where: x, y, and z are functions of t where t represents a range of real numbers. X’, y’, and z’ are the coordinates of a point on the line segment. a, b, and c define the slope of the line, such that the vector v<a, b, c> is parallel to the line. We can therefore write the parametric equation of a line segment using a t parameter that ranges between two real number values t0, t1 and a unit vector v that is in the direction of the line as follows: P = P’ + t * v

Another example is a circle. The parametric equation of the circle on the xy-plane with a center at the origin (0,0) and an angle parameter t ranging between 0 and 2π radians is: x = r cos(t) y = r sin(t) We can derive the general equation of a circle for the parametric one as follows: x/r = cos(t) y/r = sin(t) And since: cos(t)² + sin(t)² = 1 (Pythagorean identity) Then: (x/r)² + (y/r)² = 1, or x² + y² = r²

Parametric curves

Curve parameter

A parameter on a curve represents the address of a point on that curve. As mentioned before, you can think of a parametric curve as a path traveled between two points in a certain amount of time, traveling at a fixed or variable speed. If traveling takes T amount of time, then the parameter t represents a time within T that translates to a location (point) on the curve.

If you have a straight path (line segment) between the two points A and B, and v were a vector from A to B (v = B - A), then you can use the parametric line equation to find all points M between A and B as follows:

M = A + t*(B-A)

Where:

t is a value between 0 and 1.

The range of t values, 0 to 1 in this case, is referred to as the curve domain or interval. If t was a value outside the domain (less than 0 or more than 1), then the resulting point M will be outside the linear curve AB.

Figure 25: Parametric line in 3-D space and parameter interval.

The same principle applies for any parametric curve. Any point on the curve can be calculated using the parameter t within the interval or domain of values that define the limits of the curve. The start parameter of the domain is usually referred to as t0 and the end of the domain as t1.

Figure 26: Curve in 3-D space and its domain in parameter space.

Curve domain or interval

A curve domain or interval is defined as the range of parameters that evaluate into a point within that curve. The domain is usually described with two real numbers defining the domain limits expressed in the form (min to max) or (min, max). The domain limits can be any two values that may or may not be related to the actual length of the curve. In an increasing domain, the domain min parameter evaluates to the start point of the curve and the domain max evaluates to the end point of the curve.

Figure 27: Curve domain or interval is a set of two numbers that is usually ascending. When possible, domain length is set to be close to the 3d curve length, but it can be set to any length without changing the 3d curve.

Changing a curve domain is referred to as the process of reparameterizing the curve. For example, it is very common to change the domain to be (0 to 1).

Reparameterizing a curve does not affect the shape of the 3-D curve. It is like changing the travel time on a path by running instead of walking, which does not change the shape of the path.

Figure 28: Curve domain can be normalized (set to 0 to 1). Note that if the 3d curve length is much bigger than the domain length (by a factor of 10 or more), the evaluation of a parameter might not yield very accurate location on the 3d curve.

An increasing domain means that the minimum value of the domain points to the start of the curve. Domains are typically increasing, but not always.

Curve evaluation

We learned that a curve interval is the range of all parameter values that evaluate to points within the 3-D curve. There is, however, no guarantee that evaluating at the middle of the domain, for example, will give a point that is in the middle of the curve.

We can think of uniform parameterization of a curve as traveling a path with constant speed. A degree-1 line between two points is one example where equal intervals or parameters translate into equal intervals of arc length on the line as in figure (29). In parametric curves, it is rare that equal intervals of parameters evaluate to equal intervals on the 3-D curve.

Figure 29: A special case of a degree-1 line where equal parameter intervals, evaluate to equal curve lengths.

It is more likely that the speed decreases or increases along the path. For example, if it takes 30 minutes to travel a road, it is unlikely that you will be exactly half way through at minute 15. Figure (30) shows a typical case where equal parameter intervals evaluate to variable length on the 3-D curve.

Figure 30: Equal parameter intervals do not usually translate into equal distances on a parametric curves such as NURBS curves.

You may need to evaluate points on a 3-D curve that are at a defined percentage of the curve length. For example, you might need to divide the curve into equal lengths. Typically, 3-D modelers provide tools to evaluate curves relative to arc length.

Tangent vector to a curve

A tangent to a curve at any parameter (or point on a curve) is the vector that touches the curve at that point, but does not cross over. The slope of the tangent vector equals the slope of the curve at the same point. The following example evaluates the tangent to a curve at two different parameters.

Figure 31: Tangents to a curve.

Cubic polynomial curves

Hermite and Bézier curves are two examples of cubic polynomial curves that are determined by four parameters. A Hermite curve is determined by two end points and two tangent vectors at these points, while a Bézier curve is defined by four points. While they differ mathematically, they share similar characteristics and limitations.

Figure 32: Cubic polynomial curves. The Bézier curve (left) is defined by four points. The Hermite curve (right) is defined by two points and two tangents.

In most cases, curves are made out of multiple segments. This requires making what is called a piecewise cubic curve. Here is an illustration of a piecewise Bézier curve that uses 7 storage points to create a two-segment cubic curve. Note that although the final curve is joined, it does not look smooth or continuous.

Figure 33: Two Bezier spans share one point.

Although Hermite curves use the same number of parameters as Bézier curves (four parameters to define one curve), they offer the additional information of the tangent curve that can also be shared with the next piece to create a smoother looking curve with less total storage, as shown in the following.

Figure 34: Two Hermite spans share one point and a tangent.

The non-uniform rational B-spline (NURBS) is a powerful curve representation that maintains even smoother and more continuous curves. Segments share more control points to achieve even smoother curves with less storage.

Figure 35: Two degree-3 NURBS spans share three control points.

NURBS curves and surfaces are the main mathematical representation used by Rhino to represent geometry. NURBS curve characteristics and components will be covered with some detail later in this chapter.

Evaluating cubic Bézier curves

Named after its inventor, Paul de Casteljau, the de Casteljau algorithm evaluates Bézier curves using a recursive method. The algorithm steps can be summarized as follows:

Input
Four points A, B, C, D define a curve t, is any parameter within curve domain.
Output
Point R on curve that is at parameter t.
Solution
1. Find point M at t parameter on line AB. 2. Find point N at t parameter on line BC. 3. Find point O at t parameter on line CD. 4. Find point P at t parameter on line MN. 5. Find point Q at t parameter on line NO. 6. Find point R at t parameter on line PQ.

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.

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Difference between revisions of "Essential Mathematics for Computational Design"

Revision as of 18:33, 18 December 2021

Contents

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

Exploded box

Tangent spheres

Matrices and Transformations

Matrix operations

Matrix multiplication

Identity matrix

Transformation operations

Translation (move) transformation

Rotation transformation

Scale transformation

Shear transformation

Mirror or reflection transformation

Planar Projection transformation

Tutorial

Multiple transformations

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