# The lines in the plane and the points on the line at infinity

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Related KMR-pages:

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The interactive simulations on this page can be navigated with the “the Fee Viewer of the Graphing Calculator“.

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Representation and Reconstruction of a Presentant against a certain Background

Representation: $\, [ \, p_{resentant} \, ]_{B_{ackground}} \, \mapsto \, \left< \, r_{epresentant} \, \right>_{B_{ackground}}$

Reconstruktion: $\, \left( \, \left< \, r_{epresentant} \, \right>_{B_{ackground}} \, \right)_{B_{ackground}} \mapsto \,\, p_{resentant}$

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Two perpendicular lines through the point (0, m) expressed in “slope form”:

$\, [ \, l_{ine1} \, ]_{S_{lopeForm}} \, \mapsto \, \left< \; y = kx + m \; \right>_{S_{lopeForm}}$.

This form can represent all lines in the plane except the linje $\, x = 0 \,$,
which corresponds to an infinite value of the slope $\, k$.

The perpendicular line through the point (0, m) in slope form is given by

$\, [ \, l_{ine2} \, ]_{S_{lopeForm}} \, \mapsto \, \left< \; y = (-\frac{1}{k})x + m \; \right>_{S_{lopeForm}}$.

This form can represent all lines in the plane except the linje $\, y = 0 \,$,
which corresponds to an infinite value of the slope $\, k$.

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Line in two-point form

This form can represent all lines in the plane.

Let $\, A \,$ och $\, B \,$ be two “dragable” points. In the Graphing Calculator a dragable point is represented by a complex number. The $\, xy$-coordinates of the points $\, A \,$ och $\, B \,$ are therfore given by $\, (\mathrm{Re} \, A, \mathrm{Im} \, A) \,$ and $\, (\mathrm{Re} \, B, \mathrm{Im} \, B)$ respectively.

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Two perpendicular lines through the origin in angular form

This form can represent all lines (in the plane) that pass through the origin.

$\, [ \, l_{inje1} \, ]_{A_{ngularForm}} \, \mapsto \, \left< \; (x-a) \sin \phi - (y-b) \cos \phi = 0 \; \right>_{A_{ngularForm}}$.

The perpendicular line through the point genom punkten $\, (a, b) \,$ in angular form is given by
$\, [ \, l_{inje2} \, ]_{A_{ngularForm}} \, \mapsto \, \left< \; (x-a) \cos \phi + (y-b) \sin \phi = 0 \; \right>_{A_{ngularForm}}$.

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Two perpendicular lines through the origin in hyperplane form

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Two perpendicular lines through the point $\, (a, b) \,$ in “affine-hyperplane form”

This form can represent all lines in the plane.

TERMINOLOGY:An affine hyperplane is a hyperplane that may be translated so that it does not necessarily pas through the origin – in contrast to a hyperplane which always passes through the origin.

In the Graphing Calculator a dragable point is represented by a complex number.

Introduce the variables $\, a = \mathrm{Re} \, A \,$ and $\, b = \mathrm{Im} \, A \,$ and let $\, A \,$ be a dragable point.
Then the $\, xy$-coordinates of the point A $\, A \,$ are given by $\, (A_x, A_y) = (a, b)$.

$\, [ \, l_{inje1} \, ]_{A_{ffineHyperPlaneForm}} \, \mapsto \, \left< \; p (x-a) + q (y-b) = 0 \; \right>_{A_{ffineHyperPlaneForm}}$.

Then the perpendicular line through the point $\, (a, b) \,$ is given by:

$\, [ \, l_{inje2} \, ]_{A_{ffineHyperPlaneForm}} \, \mapsto \, \left< \; q (x-a) - p (y-b) = 0 \; \right>_{A_{ffineHyperPlaneForm}}$.

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Line through the point (Re A, Im A) perpendicular to the direction (Re B, Im B)

This form can represent all lines in the plane.

Let $\, A \,$ and $\, B \,$ be two dragable points. In the Graphing Calculator a dragable point is represented by a complex number. The $\, xy$-coordinates of the points $\, A \,$ and $\, B \,$ are therefore given by $\, (A_x, A_y) = (\mathrm{Re} \, A, \mathrm{Im} \, A) \,$ respectively $\, (B_x, B_y) = (\mathrm{Re} \, B, \mathrm{Im} \, B)$.

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Linje through the point (Re A, Im A) along the direction (Re B, Im B) in parameter form

This form can represent all lines in the plane.

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Euclidean points and lines in the plane are expressed in homogeneous coordinates

The Euclidean plane can be identified with the plane $\, z = 1 \,$ in the Euclidean 3D-space.

This homogeneous form of coordinates can represent all Euclidean points and lines in the plane. Moreover, it also has exactly the necessary “space” needed to be able to represent a number of new points (called points “at infinity” “). All points at infinity have their $\, z$-coordinate equal to zero and must therefore lie on one and the same line (called “the line at infinity”). This line is represented by the 3D-plane $\, z = 0$.

This form of representation leads to so-called projective geometry.

The intersection of the line at infinity with the $\, x$-axis has has the $\, xy$-coordinates $\, (a,0)$.
The intersection of the line at infinity with the $\, y$-axis has the $\, xy$-coordinates $\, (0,b)$.

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$\, \begin{pmatrix} a \\ 0 \\ 1 \end{pmatrix} \,$ , $\, \begin{pmatrix} 0 \\ b \\ 1 \end{pmatrix} \,$

$\, \lgroup \begin{pmatrix} a \\ 0 \\ 1 \end{pmatrix} \times \begin{pmatrix} 0 \\ b \\ 1 \end{pmatrix} \rgroup \cdot \begin{pmatrix} x \\ y \\ z \end{pmatrix} \, = \, 0 \,$

$\, bx + ay - abz = 0$.

$\, z = 0$.

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