DraftTin / Cube

Cube

Math

Rotation Matrix: For a 3-dimensional point, we have three angles to consider - $\alpha, \beta, \theta $, which represent the angles of rotation along the x, y, z axes.

So we have a vector {x, y, z}. To rotate it, we need to mutiply the 3D Rotation Matrix with it.

1. $\alpha$ for x axis

$$ \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos{\alpha} & -\sin{\alpha} \\ 0 & \sin{\alpha} & \cos{\alpha} \\ \end{bmatrix} $$

2. $\beta$ for y-axis

$$ \begin{bmatrix} \cos{\beta} & 0 & \sin{\beta} \\ 0 & 1 & 0 \\ -\sin{\beta} & 0 & \cos{\beta} \\ \end{bmatrix} $$

3. $\theta$ for z-axis

$$ \begin{bmatrix} \cos{\theta} & -\sin{\theta} & 0 \\ \sin{\theta} & \cos{\theta} & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} $$

Note: we can use these 3 matrices to calculate {x', y', z'} after rotation.

How to contruct 6 surfaces for a cube

We just need to construct the first surface. Then the other 5 surfaces can be easily calculated by rotating the first surface. For example, the second surface are the rotation of 90 degrees of the first surface along the x or y axis.

How to display the cube on the screen

Perspective Projection

1. We can imagine a virtual camera in front of the x-y axis plane, and set depth = 1 / z, z = z + distanceFromCam to simulate how we see the cube in the real world.

2. How to map 3-dimensional points to 2-dimensional points: According to Perspective Projection, x' = x * depth, y' = y * depth. In the program, we also need to adjust the position and size of the cube:

xp := width / 2 + horizontalOffset + K1 * ooz * x * 2
yp := height / 2 + K1 * ooz * y

3. When one depth is less that the other depth, the bigger one can cover the smaller one.