Why a Shadow Can't Tell You Depth

Casting a shadow onto a wall is a linear map that flattens the plane onto a line. Everything along the light's direction lands on the same spot, so that direction is the map's null space, and the wall line is its column space. One dimension in, one lost: rank 1, and no way to recover depth from the shadow alone.

How it works

Null space, column space and rank

In one sentence

For $A$ ( $m \times n$ ), $\operatorname{Nul} A = \{\mathbf{x} : A\mathbf{x} = \mathbf{0}\}$ and $\operatorname{Col} A = \operatorname{span}$ of its columns; their dimensions satisfy $\operatorname{rank} A + \dim\operatorname{Nul} A = n$ (rank–nullity).

General formula

{\htmlData{sym=0}{P}}{\htmlData{sym=1}{\mathbf{x}}} = \frac{{\htmlData{sym=4}{\mathbf{n}}}\cdot{\htmlData{sym=1}{\mathbf{x}}}}{{\htmlData{sym=4}{\mathbf{n}}}\cdot{\htmlData{sym=2}{\boldsymbol{\ell}}}}\,{\htmlData{sym=2}{\boldsymbol{\ell}}}, \qquad \operatorname{Col} {\htmlData{sym=0}{P}} = \operatorname{span}\{{\htmlData{sym=2}{\boldsymbol{\ell}}}\}, \quad \operatorname{Nul} {\htmlData{sym=0}{P}} = \operatorname{span}\{{\htmlData{sym=3}{\mathbf{d}}}\}, \quad \operatorname{rank} {\htmlData{sym=0}{P}} = 1

The null space is what the map can't see; the column space is everything it can produce. Whatever dimensions it loses to the first, it can't output in the second.

What the symbols mean

$P$
Projection
The shadow map: plane in, line out.
$\mathbf{x}$
Point
An object in the plane.
$\boldsymbol{\ell}$
Wall direction
The line the shadow falls on: the column space.
$\mathbf{d}$
Light direction
Points along it all cast the same shadow: the null space.
$\mathbf{n}$
Normal
Perpendicular to the light direction.

Light travels along $\mathbf{d}$ ; the shadow lands on the line through the origin along $\boldsymbol{\ell}$ (the wall).
Each point $\mathbf{x}$ slides along $\mathbf{d}$ until it hits the wall: that's a linear map $P$ , with $\mathbf{n}$ perpendicular to $\mathbf{d}$ .
Every image is a multiple of $\boldsymbol{\ell}$ , so the column space is that line: rank 1.
Every point along $\mathbf{d}$ maps to zero: the null space. By rank–nullity, $1 + 1 = 2$ dimensions in total.

Learn the math

Solution Sets of Linear SystemsLin Alg · Lesson 5, Linear Equations in Linear Algebra Subspaces of RⁿLin Alg · Lesson 13, Matrix Algebra Dimension and RankLin Alg · Lesson 14, Matrix Algebra Null Spaces, Column Spaces and Linear TransformationsLin Alg · Lesson 20, Vector Spaces

Why a Shadow Can't Tell You Depth