Distance Command

Distance( <Point>, <Object> ): Yields the shortest distance between a point and an object.

Distance((2, 1), x^2 + (y - 1)^2 = 1) yields 1
Distance((2, 1, 2), (1, 3, 0)) yields 3
Let f be a function and A be a point. Distance(A, f) yields the distance between A and (x(A), f(x(A))), that is the distance between point A and ClosestPoint(f, A).

The command works for points, segments, lines, conics, functions, and implicit curves. For functions, it uses a numerical algorithm which works better for polynomials.

Distance( <Line>, <Line> ): Yields the distance between two lines.

Distance(y = x + 3, y = x + 1) yields 1.41
Distance(y = 3x + 1, y = x + 1) yields 0
Let a: X = (-4, 0, 0) + λ*(4, 3, 0) and b: X = (0, 0, 0) + λ*(0.8, 0.6, 0). Distance(a, b) yields 2.4

The distance between intersecting lines is 0. Thus, this command is only interesting for parallel lines.

Distance( <Plane>, <Plane> ): Yields the distance between the two planes.

Let eq1: x + y + 2z = 1 and eq2: 2x + 2y + 4z = -2. Distance(eq1, eq2) yields 0.82

The distance between intersecting planes is 0. Thus, this command is only meaningful for parallel planes.