Perspective: The Geometry of Seeing
Artists use vanishing points. The camera model explains them with a rotation matrix and one division. Drag the view. The matrix and the vanishing points update live.
The matrix behind the picture
The camera turns the world with a rotation matrix, then divides by depth. A world point $\mathbf{p}$ becomes a camera point $\mathbf{p}_c$, then an image point $(u, v)$:
$$ \mathbf{p}_c = R_x(\varphi)\,R_y(\theta)\,(\mathbf{p} - \mathbf{c}), \qquad u = f\,\frac{x_c}{z_c}, \qquad v = f\,\frac{y_c}{z_c} $$
The division by $z_c$ is the whole trick. Far objects have a large $z_c$, so they project small. That single division makes railroad tracks converge.
A vanishing point is the image of a direction. Move along a direction $\mathbf{d}$ without limit. The projected point goes to:
$$ u_{\text{vp}} = f\,\frac{(R\,\mathbf{d})_x}{(R\,\mathbf{d})_z}, \qquad v_{\text{vp}} = f\,\frac{(R\,\mathbf{d})_y}{(R\,\mathbf{d})_z} $$
The position of the line drops out. Only the direction stays. As a result, all edges with the same direction meet at the same vanishing point. When $(R\,\mathbf{d})_z = 0$, the direction is parallel to the image plane. Its vanishing point is at infinity, and its edges draw as parallel lines. The robot-arm notes use these same rotation matrices.
TRY_IT // PROJECT_A_BOX
Edit the code. Press RUN.What the artist counts
Look straight down the street. Two axes stay parallel to the canvas. Only the depth axis converges. One vanishing point sits on the horizon.
Turn to face a corner. Both horizontal axes converge. Two vanishing points sit on the horizon. Vertical edges stay vertical.
Tilt the camera up or down. The vertical axis also converges. The third vanishing point leaves the horizon. Towers taper toward the sky.
FIG // DRAFTING_VS_SEEING
Move the slider: camera walks away, lens zooms inPerspective in CAD and motion studies
A CAD drawing must carry true dimensions. The perspective divide changes a length with its depth, so drafting removes the divide. Move the camera far away, and zoom in so the part keeps its size. The ratio $f/z_c$ then goes to one constant $s$ for every point:
$$ u = f\,\frac{x_c}{z_c} \;\longrightarrow\; u = s\,x_c \qquad (z_c \to \infty) $$
This limit is orthographic projection. Parallel edges stay parallel. There are no vanishing points. The front, top, and side views on a drawing are parallel projections along the three axes. The isometric view is a parallel projection along the cube diagonal, with the part turned $45^\circ$ and tipped $35.26^\circ$, the pose in the figure above.
Motion studies go the other way. A machine-vision camera obeys the full perspective model on this page. A robot finds the pose of a part when it inverts these equations, with the same rotation matrices as our Machine Design course and the inverse-kinematics simulation. The artist, the drafter, and the robot all use one geometry. They keep or drop the divide by $z$.