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# Geometric Primitives Notes

This is as part of Mathematics studying for Graphics and Game development notes (in series).

## Representation Techniques

It can be represented via

1. Implicit Form

For example,

$x^2 + y^2 + z^2 = 1$

This equation is true for all points on the surface of a unit sphere centered at the origin.

2. Parametric Form

For example,

$x(t) = \cos 2{\pi}t$ $y(t) = \sin 2{\pi}t$

3. Straightforward Form

Capture the most important and obvious information directly. For example, line segment, it has 2 endpoints. For a sphere, it has center and radius.

4. Degrees of Freedom

Each one has its own degrees of freedom. It’s a minimum number of required information to describe such geometric primitive. Often we will find redundant piece of information that we could eliminate i.e. vector having unit length.

## Line and Rays

### Parametric Representation of Rays

For 2D ray can be defined with the following parametric equations

$x(t) = x_0 + t{\Delta}x$ $y(t) = y_0 + t{\Delta}y$

Similarly to 3D as we just add another equation for another dimension to it.

To represent it in vector notation,

$p(t) = p_0 + td$

### Special 2D Representation of Lines

In 2D we can represent line with

$ax + by = d$

Note: Other resource might use another form of (7) as $ax + by + d = 0$ This flips the sign of $$d$$.

Alternatively, we can represent above equation with dot product as follows

$p \cdot n = d$

Look closely, above two equations are in similar form of normal line equation as follows

$y = mx + b$

$$m$$ is slope and called rise over run.

Notes for (9)

• $$n$$ which is unit vector gives the signed distance from the origin to the line. $$d$$ is positive if the line moves in the direction of $$n$$.

## Spheres and Circles

The sphere equation comes from a simple definition of set of all points that are a given distance from the center. Thus it comes down to

${\Vert}p-c{\Vert} = r$

where $$p$$ is any point on the surface of the sphere, $$c$$ is the center of the sphere, and $$r$$ is the radius.

If we expand the above equation, then we get the form we get used to as follows.

$(x-c_x)^2 + (y-c_y)^2 + (z-c_z)^2 = r^2$

## Bounding Box

### Representing AABBs

It’s based on keeping track of minimum and maximum value to represent the axis-aligned bounding box.

$P_{min} = [\ x_{min}\ \ y_{min}\ \ z_{min}\ ]$ $P_{max} = [\ x_{max}\ \ y_{max}\ \ z_{max}\ ]$

The center point $$c$$ is given by

$c = ( P_{min} + P_{max})/2$

The size vector $$s$$ is the vector from $$P_{min}$$ to $$P_{max}$$ in turns this contains width, height, and length of the box.

We can get its radius by

\begin{align} r &= P_{min} - c \\ &= s/2 \end{align}

## Planes

### Implicit Definition - The Plane Equation

$ax + by + cz = d$ $p \cdot n = d$

### Defining Using Three Points

Assume there are two edges. $$e_1$$ is $$\overrightarrow {p_{2}p_{3}}$$, and $$e_3$$ is $$\overrightarrow {p_{2}p_{1}}$$ in which those 3 points are in the plane.

$e_3 = p_2 - p_1$ $e_1 = p_3 - p_2$

### Distance from Point to Plane

Distance from point $$p$$ on the plane to point $$q$$ closest to it (which is not on the plane) is

\begin{align} p + an &= q \nonumber \\ (p + an) \cdot n &= q \cdot n \nonumber \\ p \cdot n + (an) \cdot n &= q \cdot n \nonumber \\ d + a &= q \cdot n \nonumber \\ a &= q \cdot n -d \end{align}

where $$a$$ is the amount along $$\vec n$$.

## Triangles

Law of sines (see proof)

$\frac{\sin{\theta}_1}{l_1} = \frac{\sin{\theta}_2}{l_2} = \frac{\sin{\theta}_3}{l_3}$

Law of cosines (see proof)

$l_1^2 = l_2^2 + l_3^2 - 2l_2l_3\cos\theta_1$ $l_2^2 = l_1^2 + l_3^2 - 2l_1l_3\cos\theta_2$ $l_3^2 = l_1^2 + l_2^2 - 2l_1l_2\cos\theta_3$

Perimeter is

$p = l_1 + l_2 + l_3$

### Barycentric Space

Any point inside a triangle can be expressed as the weighted average of the vertices. The weights are known as barycentric coordinate.

Conversion from barycentric coordinates to 3D position

$(b_1, b_2, b_3) \iff b_1v_1 + b_2v_2 + b_3v_3$

where $$b_1 + b_2 + b_3 = 1$$

We can interpret it as ratios of areas

$b_1 = A(T_1)/A(T)$ $b_2 = A(T_2)/A(T)$ $b_3 = A(T_3)/A(T)$

## Resource

• 3D Math Primer for Graphics and Game development - Chapter 12

First published on Aug, 4, 2019

Written by Wasin Thonkaew
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