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Lab · c08.s02.L1

Projection shadow

The closest point in a subspace is the one whose leftover is perpendicular to it.

Question: What is the closest point in a subspace?

Projection shadowA rust-red arrow rises from the origin of a tilted plane. A dashed navy line drops from the arrow's tip to a navy dot on the plane, meeting it at a right angle, so that dot is the closest point of the plane. As the arrow swings the dot slides round a dashed circle; raising the arrow lengthens the drop. A gold rival point, on a dotted line through the dot, is never nearer.Projection shadowClosest point = foot of the perpendicular dropu₁u₂arrow from the origindrop = distancerival pointplane through the origin (u₁, u₂)azimuth 60° · height 0.90distance 0.900rival offset +0.60rival distance 1.082
Schematic Schematic: the arrow, its shadow and the gold rival are a construction drawn on a fixed camera to explain projection, not a measurement of anything. Every coordinate, distance and angle shown is computed by data/scripts/c08.s02.L1.py.
Lab controls
60 degrees

Left/Right move one 15° stop; Page Up/Down move 90°; Home 0°; End 345°

0.9 height

Left/Right move 0.10; Page Up/Down move 0.50; Home 0.20; End 1.60

0.6 offset

Left/Right move 0.10; Page Up/Down move 0.50; Home −1.40; End 1.40

Frame 1 of 10: The straight dropThe lab opens on its default: the arrow points to azimuth 60° and rises to height 0.90, and the gold rival sits at offset +0.60. The dashed drop falls from the tip to the navy dot and the small right-angle mark records that the drop meets the plane square-on, so that dot is the closest point the plane offers. The drop is the distance, 0.900; the rival is 1.082 away, farther than the shadow.

Keyboard: Space plays or pauses, Left and Right step, Home goes to the first frame, End goes to the last frame.

Text description of this lab

A rust-red arrow rises from the origin of a tilted plane. A dashed navy line drops from the arrow's tip to a navy dot on the plane, meeting it at a right angle, so that dot is the closest point of the plane. As the arrow swings the dot slides round a dashed circle; raising the arrow lengthens the drop. A gold rival point, on a dotted line through the dot, is never nearer.

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Caption status: schematic. Every lab declares one of published, re-simulated, schematic, synthetic or interpolated.