Exercise 3: Why Precomputing a Combined Transform Matters for 10,000 Vertices — Possible Solution
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WHAT THE TWO APPROACHES ACTUALLY COST
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Transforming each of 10,000 vertices through local -> world -> camera
as three SEPARATE matrix multiplications means performing three full
matrix-vector multiplications per vertex - 30,000 matrix-vector
multiplications total for the whole object, every single frame.
Precomputing the single combined local-to-camera matrix ONCE (by
multiplying the world-to-camera matrix and the local-to-world matrix
together, a single matrix-matrix multiplication done once per object,
not per vertex) and then applying that one combined matrix to each
vertex means only 10,000 matrix-vector multiplications total - one
per vertex, instead of three.

WHY THIS SAVES REAL WORK, NOT JUST CODE COMPLEXITY
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This chapter verified directly that applying each transformation step
separately and applying one precomputed combined matrix produce
EXACTLY the same result (a measured difference of 0.0 in the worked
pipeline example) - so there is no accuracy tradeoff at all, only a
work tradeoff. Since every one of the object's 10,000 vertices needs
the identical local-to-world and world-to-camera transformation
(the object as a whole has only one position and orientation, shared
by every vertex belonging to it), computing that combined
transformation once and reusing it for every vertex avoids redoing
the exact same matrix-matrix multiplication 10,000 times over for no
benefit - the local-to-world and world-to-camera matrices don't
change from one vertex to the next within the same object and the
same frame.

THE SCALE OF THE SAVINGS
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Precomputing the combined matrix reduces the total number of matrix
multiplications from roughly three times the vertex count (30,000) to
the vertex count itself plus one extra matrix-matrix multiplication
(10,000 + 1), which is very close to a threefold reduction in the
total transformation work for that object, at essentially zero
implementation cost (a small amount of extra code to multiply the two
transform matrices together once) and, per this chapter's own
verified finding, zero cost in accuracy.

WHY THIS WORKS AS AN ANSWER
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The answer quantifies the actual number of matrix operations each
approach requires rather than describing the improvement only
qualitatively, explains why every vertex of the same object shares
the identical local-to-world and world-to-camera matrices (making the
precomputation valid), and grounds the claim that no accuracy is lost
in this chapter's own explicitly verified 0.0-difference finding.