Midpoint

The point halfway between two values.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageMidpoint!↗

Midpoint(a, b)The point halfway between two values.

Typed signatures

Type	Package	Applies
(any, any) -> unknown	structures	always

Domain: Structures

Details

• The AffineMidpoint protocol's member: numbers, complex numbers, and lists componentwise, so vectors, matrices and Points have midpoints too. Two lists of different shapes have none.
• Round uses it through MidpointOrder (ticks plus midpoints), comparing a value against the midpoint between the ticks either side of it.
• Mathlib's midpoint, defined on an affine space over a ring where 2 is invertible, and Wolfram's Midpoint of the segment between a and b. The protocols themselves are our extension: Mathlib has no class for either.

Examples

Basic #

Between two numbers

live ↗

In$\mathrm{Midpoint}(1, 2)$Out$\frac{3}{2}$

Between two points given by their coordinates, componentwise

live ↗

In$\mathrm{Midpoint}(\bigl\lbrack1, 1\bigr\rbrack, \bigl\lbrack2, 3\bigr\rbrack)$Out$\bigl\lbrack\frac{3}{2}, 2\bigr\rbrack$

In any dimension

live ↗

In$\mathrm{Midpoint}(\bigl\lbrack1, 2, 3\bigr\rbrack, \bigl\lbrack4, 5, 6\bigr\rbrack)$Out$\bigl\lbrack\frac{5}{2}, \frac{7}{2}, \frac{9}{2}\bigr\rbrack$

A Point keeps its head

live ↗

In$\mathrm{Midpoint}(\mathrm{Point}(\bigl\lbrack1, 1\bigr\rbrack), \mathrm{Point}(\bigl\lbrack2, 3\bigr\rbrack))$Out$\mathrm{Point}(\bigl\lbrack\frac{3}{2}, 2\bigr\rbrack)$

Scope #

Midpoint of two points, exact: a half stays rational

live ↗

In$\mathrm{Midpoint}(\bigl\lbrack1, 1\bigr\rbrack, \bigl\lbrack3, 4\bigr\rbrack)$Out$\bigl\lbrack2, \frac{5}{2}\bigr\rbrack$

Works in any dimension, coordinate by coordinate

live ↗

In$\mathrm{Midpoint}(\bigl\lbrack1, 2, -1, 3\bigr\rbrack, \bigl\lbrack4, 2, -3, -2\bigr\rbrack)$Out$\bigl\lbrack\frac{5}{2}, 2, -2, \frac{1}{2}\bigr\rbrack$

Point wrappers are kept on the result

live ↗

In$\mathrm{Midpoint}(\mathrm{Point}(\bigl\lbrack1, 2\bigr\rbrack), \mathrm{Point}(\bigl\lbrack-2, 3\bigr\rbrack))$Out$\mathrm{Point}(\bigl\lbrack-\frac{1}{2}, \frac{5}{2}\bigr\rbrack)$

live ↗

In$\mathrm{Midpoint}(\bigl\lbrack a, b\bigr\rbrack, \bigl\lbrack c, d\bigr\rbrack)$Out$\bigl\lbrack\mathrm{Midpoint}(a, c), \mathrm{Midpoint}(b, d)\bigr\rbrack$

Properties #

Entry by entry, since a matrix is a list of vectors

live ↗

In$\mathrm{Midpoint}(\bigl\lbrack\bigl\lbrack1, 1\bigr\rbrack, \bigl\lbrack0, 2\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack3, 3\bigr\rbrack, \bigl\lbrack4, 0\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack2, 2\bigr\rbrack, \bigl\lbrack2, 1\bigr\rbrack\bigr\rbrack$

In the complex plane

live ↗

In$\mathrm{Midpoint}(1+2i, 3)$Out$2+i$

Averaging each coordinate: $(1-3)/2$ and $(2+5)/2$

live ↗

In$\mathrm{Midpoint}(\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack-3, 5\bigr\rbrack)$Out$\bigl\lbrack-1, \frac{7}{2}\bigr\rbrack$

From the origin, the midpoint is half the other point

live ↗

In$\mathrm{Midpoint}(\bigl\lbrack0, 0\bigr\rbrack, \bigl\lbrack1, 3\bigr\rbrack)$Out$\bigl\lbrack\frac{1}{2}, \frac{3}{2}\bigr\rbrack$

Possible issues #

Integer coordinates come out integer when the sums are even

live ↗

In$\mathrm{Midpoint}(\mathrm{Point}(\bigl\lbrack-1, 2\bigr\rbrack), \mathrm{Point}(\bigl\lbrack3, -2\bigr\rbrack))$Out$\mathrm{Point}(\bigl\lbrack1, 0\bigr\rbrack)$

About

Examples	13
Checked against	Wolfram Language 9 agree3 differ1 unevaluated
Crosswalk	Wolfram Language

Implementation

mappedwolframpackages/oracle/src/mappings.ts

Wolfram's takes the segment as one list of its two ends.

See also: LowerTick, UpperTick, Round