# InfiniteLine

InfiniteLine[{p1,p2}]

represents the infinite straight line passing through the points p1 and p2.

InfiniteLine[p,v]

represents the infinite straight line passing through the point p in the direction v.

# Details • InfiniteLine is also known as line.
• InfiniteLine can be used as a geometric region and graphics primitive.
• • InfiniteLine represents linear curve or .
• InfiniteLine can be used in Graphics and Graphics3D.
• InfiniteLine will be clipped by PlotRange when rendering.
• In graphics, the points p, pi and vector v can be Dynamic expressions.
• Graphics rendering is affected by directives such as Thickness, Dashing, and color.
• InfiniteLine can be used with symbolic points in GeometricScene.

# Examples

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## Basic Examples(3)

An InfiniteLine in 2D:

And in 3D:

Different styles applied to an infinite line:

Test point membership in an infinite line:

## Scope(19)

### Graphics(8)

#### Specification(3)

Define an InfiniteLine containing and going in the direction :

Define the same line passing through and :

Define a 3D infinite line containing and going in the direction :

Define the same infinite line using the points and :

An infinite line varying direction:

#### Styling(4)

Thickness in scaled size:

Thickness in printer's points:

Infinite lines can be rendered in dashed or dotted styles:

Color directives can be used to specify the color of an infinite line:

Combine various directives to style an infinite line:

#### Coordinates(1)

Points and vectors can be Dynamic:

### Regions(11)

Embedding dimension is the dimensionality of the vertices:

Geometric dimension is the dimension of the region:

Point membership test:

Get conditions for membership:

An infinite line is unbounded:

Find the region range:

InfiniteLine has infinite measure:

Distance from a point:

Plotting distance to the region:

Signed distance from a point:

Distance to the nearest point:

Nearest point in the region:

Visualizing nearest points:

Integrate over an infinite line:

Optimize over an infinite line:

Solve equations on an infinite line:

## Applications(17)

Create parallel lines aligned to :

Illustrate asymptotes:

Convert the intercept form of a line to an InfiniteLine:

Visualize lines:

Convert the point slope form of a line to an InfiniteLine:

Visualize lines:

Convert the slope intercept form of a line to an InfiniteLine:

Visualize lines:

Convert the two-point form of a line to an InfiniteLine:

Visualize lines:

Convert the parametric form of a line to an InfiniteLine:

Visualize lines:

The tangent line to a parametric curve f[u] is given by InfiniteLine[f[u],f'[u]]. Find the tangent line to the parametric curve :

Find the tangent line for the parametric curve :

Find the intersection of InfiniteLine[{0,0},{1,1}] and InfiniteLine[{{0,1},{1,0}}]:

Plot it:

Find the intersections of InfiniteLine[{0,0},{1,1}] and Circle[{0,0},1]:

Plot it:

Find all pairwise intersections between five random lines:

Use BooleanCountingFunction to express that exactly two conditions are true:

Plot it:

Find the intersection of InfiniteLine[{{-1,1,1},{1,1,1}}] and InfinitePlane[{{2,0,0},{0,2,0},{0,0,2}}]:

Plot it:

Find the intersections of InfiniteLine[{{-1,1,1},{1,1,1}}] and Sphere[{0,0,0},3]:

Plot it:

Find the intersections of InfiniteLine[{{-1,1/3,1/2},{1,1/3,1/2}}] and the boundary of Tetrahedron[{{0,0,0},{1,0,0},{0,1,0},{0,0,1}}]:

Plot it:

Indicate Mean on a Histogram:

Visualize the axis of rotation for RotationTransform:

Find the altitude of a triangle:

Visualize altitude in red:

## Properties & Relations(4)

InfiniteLine[{p1,p2}] is equivalent to InfiniteLine[p1,p2-p1]:

ParametricRegion can represent any InfiniteLine:

ImplicitRegion can represent any InfiniteLine:

InfiniteLine is a special case of ConicHullRegion:

## Neat Examples(2)

A random collection of lines:

Organized collection of lines: