TriangleWave

The triangle wave: period 1, default range {-1, 1}, sine-like phase (0 at x = 0, peak at x = 1/4).

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TriangleWave(x)a triangular approximation to sin(2Pix) -- 0 at x = 0, 1 at x = 1/4, 0 at x = 1/2, -1 at x = 3/4.

TriangleWave({min, max}, x)the triangle wave rescaled to vary from min to max, unit period.

Typed signatures

Type	Package	Applies
`(real \| vector<real^2>, real?) -> number`	`analytic`	always

Domain: Elementary functions

Details

Default range is {-1, 1} with a sine-like phase, NOT the [0, 1] tent UnitTriangle/HeavisideLambda use -- TriangleWave(0) = 0, not 1.
Continuous everywhere, including at the peak and trough.
The range form takes the range FIRST, x second -- TriangleWave({min, max}, x) -- matching SawtoothWave and SquareWave.

Examples

Basic #

Zero-crossing at x = 0, unlike UnitTriangle's peak there

\mathrm{TriangleWave}(0)

Out

0

The peak, at the quarter period

\mathrm{TriangleWave}(\frac{1}{4})

Out

1

Zero-crossing again at the half period

\mathrm{TriangleWave}(\frac{1}{2})

Out

0

The trough, at the three-quarter period

\mathrm{TriangleWave}(\frac{3}{4})

Out

-1

Scope #

Rescaled to a {min, max} range -- the range's midpoint at the half period

\mathrm{TriangleWave}(\bigl\lbrack0, 10\bigr\rbrack, \frac{1}{2})

Out

5

An exact irrational input stays exact through the affine branch

\mathrm{TriangleWave}(-π)

Out

12-4π

Known value · f = Frac(-π) = 4 - π ∈ (3/4, 1), so TriangleWave = 4f - 4

Properties #

Period 1

\mathrm{TriangleWave}(\frac{1}{4})=\mathrm{TriangleWave}(3\frac{1}{4})

Out

\mathrm{True}

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