# UnitTriangle

UnitTriangle[x]

represents the unit triangle function on the interval .

UnitTriangle[x1,x2,]

represents the multidimensional unit triangle function on the interval .

# Details # Examples

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## Basic Examples(4)

Evaluate numerically:

Plot in one dimension:

Plot in two dimensions:

UnitTriangle is a piecewise function:

## Scope(34)

### Numerical Evaluation(5)

Evaluate numerically:

Evaluate to high precision:

For inputs between -1 and 1, the precision of the output tracks the precision of the input:

For inputs outside that range, the result is exact:

Evaluate efficiently at high precision:

### Specific Values(4)

Values of UnitTriangle at fixed points:

Value at zero:

Evaluate symbolically:

Find a value of x for which UnitTriangle[x]=0.4:

### Visualization(4)

Plot the UnitTriangle function:

Visualize scaled UnitTriangle functions:

Visualize the composition of UnitTriangle with a periodic function:

Plot UnitTriangle in three dimensions:

### Function Properties(11)

Function domain of UnitTriangle:

It is restricted to real inputs:

Function range of UnitTriangle:

UnitTriangle is an even function:

The area of the UnitTriangle is 1:

UnitTriangle is not an analytic function:

It has singularities:

However, it is continuous everywhere:

Verify the claim at one of its singular points:

UnitTriangle is neither nondecreasing nor nonincreasing:

UnitTriangle is not injective:

UnitTriangle is not surjective:

UnitTriangle is non-negative:

UnitTriangle is neither convex nor concave:

### Differentiation and Integration(6)

First derivative with respect to x:

Higher-order derivatives with respect to x:

First derivative with respect to z:

Series expansion at the origin:

Compute the indefinite integral using Integrate:

Verify the anti-derivative away from the singular points:

Definite integral:

### Integral Transforms(4)

FourierTransform of UnitTriangle is a squared Sinc function:

Find the LaplaceTransform of UnitTriangle:

The convolution of UnitTriangle with itself:

## Applications(3)

Integrate a piecewise function involving UnitTriangle symbolically and numerically:

Solve a differential equation involving UnitBox and UnitTriangle:

Generate data from some distribution:

Apply mean shift until all data points have converged:

Gather the result into clusters:

Visualize the clustering:

## Properties & Relations(3)

The derivative of UnitTriangle is a piecewise function:

The derivative of HeavisideLambda is a distribution:

At higher orders, the DiracDelta distribution appears:

Multidimensional unit triangle function equals the product of 1D functions for each argument:

UnitTriangle is a special case of BSplineBasis: