Half Your Triangles Face Away

A closed 3D model has outward-facing front triangles and inward-facing back triangles. When the camera sits outside the mesh, back faces are never visible. They would be hidden by front faces and the z-buffer anyway.

Back-face culling skips rasterizing away-facing triangles. For closed opaque models, this eliminates roughly 50% of triangles before any per-pixel work begins. OpenGL, Vulkan, and Direct3D enable it by default.

The Cross Product Test

For triangle vertices A, B, C in counter-clockwise order (front-facing):

c
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For example, a triangle on the XY plane facing +Z with camera at (0,0,5) has N pointing toward the camera and survives. A triangle facing -Z gets culled because the normal points away. The dot product measures cosine of the angle between normal and view vector; negative means angle > 90 degrees.

CCW winding: convention for front faces in right-handed systems
Two-sided materials: disable culling or triangles vanish
Open surfaces: culling creates holes on single-sided geometry

For This Exercise

You will implement the culling test for a set of triangles and count survivors. This task requires computing face normals via cross product and comparing against the camera position. Wrong winding order when importing Blender or Maya meshes is one of the most common bugs in custom renderers.

Your Task

Decide which triangles face the camera (back-face culling) with exact 3D integer arithmetic, three ways:

World space (the correct rule): the triangle is front-facing if its normal points toward the eye, n·(eye - A) > 0, with n = (B-A) × (C-A).
After perspective projection: the sign of the projected triangle's signed area. It must agree with the world-space test for triangles in front of the camera.
The xy shortcut: the signed area of the triangle's x-y footprint, ignoring perspective. It is right for an orthographic camera but wrong for oblique triangles seen in perspective. Report where it fails.

Input

# starts a comment line.

c
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Rules

Degenerate: a zero normal prints NAME: degenerate (zero normal), discarded and counts in no total.
World test: d = n·(eye - A), computed in 64-bit. With front cw, every sign is negated. d > 0 is front, d < 0 is back, and 0 is edge-on.
Projection: x' = (x - ex)/(ez - z) and y' = (y - ey)/(ez - z). Only triangles with every z < ez can be projected; otherwise print n/a (not in front of the camera). The sign of the projected area is the sign of det[[X0, Y0, D0], [X1, Y1, D1], [X2, Y2, D2]] with X = x - ex, Y = y - ey, D = ez - z, adjusted for the winding. Append (MISMATCH) if it disagrees with the world test (it never should).
xy shortcut: the sign of (B-A).x·(C-A).y - (B-A).y·(C-A).x, adjusted for the winding. Append (wrong) when it disagrees with the world test.

Output

c
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Summary: it appends ; N not fully in front of the camera when any triangle could not be projected. With front cw, the header says clockwise.
Errors: camera: X Y Z (within +-1000), front: ccw|cw, tri: NAME AX AY AZ BX BY BZ CX CY CZ, tri: coordinates within +-1000, at most 16 triangles, unknown command X.

Example

Input:

c
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Output:

c
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Requirements

Use integer cross and dot products. With integer vertices, the classification is exact, so edge-on really means zero.
The projected-area determinant avoids any division. Multiplying through by the positive depths keeps the sign.

Hidden tests cover an oblique triangle that fools the shortcut, a triangle behind the camera, an edge-on triangle, clockwise winding, a camera looking from the other side, and input errors.

Your Task

Decide which triangles face the camera (back-face culling) with exact 3D integer arithmetic, three ways:

World space (the correct rule): the triangle is front-facing if its normal points toward the eye, n·(eye - A) > 0, with n = (B-A) × (C-A).

After perspective projection: the sign of the projected triangle's signed area. It must agree with the world-space test for triangles in front of the camera.

The xy shortcut: the signed area of the triangle's x-y footprint, ignoring perspective. It is right for an orthographic camera but wrong for oblique triangles seen in perspective. Report where it fails.

Rules

Degenerate: a zero normal prints NAME: degenerate (zero normal), discarded and counts in no total.

World test: d = n·(eye - A), computed in 64-bit. With front cw, every sign is negated. d > 0 is front, d < 0 is back, and 0 is edge-on.

Projection: x' = (x - ex)/(ez - z) and y' = (y - ey)/(ez - z). Only triangles with every z < ez can be projected; otherwise print n/a (not in front of the camera). The sign of the projected area is the sign of det[[X0, Y0, D0], [X1, Y1, D1], [X2, Y2, D2]] with X = x - ex, Y = y - ey, D = ez - z, adjusted for the winding. Append (MISMATCH) if it disagrees with the world test (it never should).

xy shortcut: the sign of (B-A).x·(C-A).y - (B-A).y·(C-A).x, adjusted for the winding. Append (wrong) when it disagrees with the world test.

Output

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Summary: it appends ; N not fully in front of the camera when any triangle could not be projected. With front cw, the header says clockwise.

Errors: camera: X Y Z (within +-1000), front: ccw|cw, tri: NAME AX AY AZ BX BY BZ CX CY CZ, tri: coordinates within +-1000, at most 16 triangles, unknown command X.

Requirements

Use integer cross and dot products. With integer vertices, the classification is exact, so edge-on really means zero.

The projected-area determinant avoids any division. Multiplying through by the positive depths keeps the sign.

Hidden tests cover an oblique triangle that fools the shortcut, a triangle behind the camera, an edge-on triangle, clockwise winding, a camera looking from the other side, and input errors.

Back-Face Culling with Cross Products

Topic Overview

Half Your Triangles Face Away

The Cross Product Test

For This Exercise

Implementation Task

Your Task

Input

Rules

Output

Example

Requirements

Solve it with a plan

Progressive Hints

Resources

Learn by building

Half Your Triangles Face Away

The Cross Product Test

For This Exercise

Your Task

Input

Rules

Output

Example

Requirements