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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)

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



First published on Aug, 4, 2019






Written by Wasin Thonkaew
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