The Perspective Box

"A face pasted flat is a mask. A face on a box is a head."

The Head Is a Box

Wrap the head in a rectangular box and something clicks. The box's faces become the planes of the head: a front where the face lives, two sides for the temples and cheeks, a top for the crown, a bottom for the jaw, a back for the skull. The box is cruder than the ball — and that crudeness is the point. A box has unmistakable faces and edges, so the instant you draw one you have committed to which way the head is turned.

Why the Box, Not Just the Ball

The ball from earlier gives you mass, but a sphere looks the same from every angle — it carries no direction. The box adds what the ball lacks: three axes and six faces, a little coordinate frame strapped to the head. From the box you can read yaw (turned left or right), pitch (nodding up or down), and roll (tilted toward a shoulder). Orientation stops being a vague feeling and becomes something with edges you can actually draw and check.

Features Ride the Planes

Once the box exists, the features stop floating. The eyes sit on the front plane and tilt with it; the ears pin to the side planes; the brow wraps around the top edge. As the box turns away, the far side compresses and the features foreshorten automatically, because they are riding a solid that is turning, not sitting on a flat sheet. Draw the box first and correct perspective is inherited rather than guessed. Skip the box and every feature is an independent gamble.

Where This Track Meets the Next

Here is the hinge of the whole quest. The box has an orientation — a rotation — and that rotation is precisely the one thing a flat photo hides. You, the artist, judge it by eye. The engine cannot eyeball; it must compute it. Recovering the box's rotation from where a face's features landed in a photo is exactly the camera-solve problem, and it is the entire subject of the next track. The human perspective box you draw and the rotation matrix Loomis solves for are the same object, reached two different ways.

Code

The head box is an oriented coordinate frame·python

# The head box is just an origin plus three axes -- a little frame on the head.
# Its orientation is the ONE thing the photo hides and the camera solve recovers.
class HeadBox:
    origin: Vec3    # center of the head, in u
    right:  Vec3    # +x : ear to ear
    up:     Vec3    # +y : chin to crown
    fwd:    Vec3    # +z : back of skull toward the nose

# Given the box rotation R and translation t, every 3D feature point
# projects to a pixel in the photo:
#     pixel = project(R @ feature_in_u + t, focal)
# Draw the box, and the eyes / nose / ears land where perspective forces them.
# Next track: given the pixels, SOLVE for R. That inversion is the camera solve.

Exercise

Draw a plain cube three ways: straight-on, in three-quarter view, and tilted as if seen from slightly above. For each cube, sketch a simple head inside it — just the front plane's eyeline and a nose direction. Watch how the same head reads as three completely different turns purely because the box changed. You didn't move the features; you moved the box.

Hint

This is the artist's version of what the engine computes. You set the box's orientation by hand and the features follow; Loomis is handed the features and has to solve backward for the box. Same object, opposite directions — which is exactly the forward/inverse split from Track 1, now made concrete.