ATRACTOR

Ellipse

Let \(F_{1}\) and \(F_{2}\) be two distinct points of the plane and \(D\) a number bigger than the distance (in the geometry we are considering) between \(F_{1}\) and \(F_{2}\). The geometric place of the points of the plane whose the sum of the distances to \(F_{1}\) and \(F_{2}\) is \( D\) is called an ellipse with foci \(F_{1}\) and \(F_{2}\).

In the taxicab geometry, an ellipse can take different shapes depending on the relative positions of the foci. Let us look at some examples ².

Consider \(\overrightarrow{u}=\overrightarrow{F_{1}F_{2}}\), with \(\overrightarrow{u}=(u_{1},u_{2})\).

If \(\overrightarrow{u}\) is horizontal or vertical (that is, \(u_{2}=0\) or \(u_{1}=0\)), the ellipse has the shape of an hexagon.

When the vector \(\overrightarrow{u}\) is neither horizontal nor vertical, the ellipse has the shape of an octagon.

In the module, it is possible to vary the position of the foci and discover the relation between the length of the sides of the hexagon/octagon, with the distance (in the taxicab geometry) between the foci.

Hyperbola

Let \(F_{1}\) and \(F_{2}\) be two distinct points of the plane and \(D\) a number smaller than the distance (in the geometry we are considering) between \(F_{1}\) and \(F_{2}\). The geometric place of the points of the plane such that the absolute value of the difference between the distances to \(F_{1}\) and \(F_{2}\) is \(D\) is called an hyperbola with foci \(F_{1}\) and \(F_{2}\).

In the taxicab geometry, an hyperbola may assume several shapes, depending of the relative positions of the foci. Let us see some examples ³.

Consider \(\overrightarrow{u}=\overrightarrow{F_{1}F_{2}}\), with \(\overrightarrow{u}=(u_{1},u_{2})\).

If \(\overrightarrow{u}\) is horizontal (that is \(u_{2}=0\)), the hyperbola is given by two vertical parallel lines.

Similarly, if the vector \(\overrightarrow{u}\) is vertical (that is \(u_{1}=0\)), the hyperbola is given by two horizontal parallel lines.

If vector \(\overrightarrow{u}\) is neither horizontal nor vertical, we can still split it into the following two cases:

• if \(|| u_{1}|-| u_{2}|| = D\), the hyperbola is the union of two unbounded regions, two line segments, that make an angle of \(45^{\circ}\) with the horizontal, and two half-lines, which are vertical if \(| u_{2}|<| u_{1}|\) and which are horizontal when \(| u_{2}|>| u_{1}|\);
• if \(|| u_{1}|-| u_{2}|| \neq D\), the hyperbola is the union of two line segments, that make an angle of \(45^{\circ}\) with the horizontal, with four half-lines: every line being vertical if \(| u_{2}|<| u_{1}|\); every line being horizontal when \(| u_{2}|>| u_{1}| \); and two horizontal lines and two vertical lines if \(| u_{2}|=| u_{1}|\).

In the applet, it is possible to change the distance/position of the foci and discover the relation between it and the length of the segments.

Parabola

Let \(d\) be a line of the plane and \(F\) be a point not belonging to \(d\). The geometric place of the points of the plane which are equidistant from \(F\) and from \(d\) is called a parabola with focus \(F\) and directrix \(d\).

In the taxicab geometry, a parabola may have several shapes, depending on the slope of the directrix. Let us see some examples.

If the directrix has slope \(1\) or \(-1\), the parabola is given by a line segment parallel to the directrix and by two half-lines (one horizontal and the other vertical).

If the absolute value of the slope of the directrix is smaller than \(1\), the parabola is given by two line segments and by two vertical half-lines.

If the absolute value of the slope of the directrix is greater than \(1\), the parabola is given by two line segments and two horizontal half-lines.

Find with the help of the applet the relationship between the slopes of the line segments, if any, with the slope of the directrix.

Bisector

Let \(A\) and \(B\) be two distinct points of the plane. We define the Bisector of \([AB]\) as the geometric place of the points of the plane that are equidistant (in the geometry we are considering) from \(A\) and \(B\).

In the taxicab geometry, a bissector may have several shapes, depending of the relative positions of points \(A\) and \(B\). Consider \(\overrightarrow{u}=\overrightarrow{AB}\), with \(\overrightarrow{u}=(u_{1},u_{2})\). Then:

• if \(\overrightarrow{u}\) is horizontal (that is, \(u_{2}=0\)), the bissector is given by a vertical line;
• similarly, if \(\overrightarrow{u}\) is vertical (that is, \(u_{1}=0\)), the bissector is given by a horizontal line.

If \(\overrightarrow{u}\) is neither horizontal nor vertical, we can distinguish the following cases:

• if \(| u_{1}|=| u_{2}|\), that is, if the line \(AB\) has slope \(1\) or \(-1\), then the bissector is formed by two unbounded regions and by a line segment, that makes an angle of \(45^{\circ}\) with the horizontal;
• if \(| u_{1}|\neq| u_{2}|\), the bissector is formed by a line segment that makes an angle of \(45^{\circ}\) with the horizontal and by two half lines, that are vertical when \(| u_{2}|<| u_{1}|\) and horizontal when \(| u_{2}|>| u_{1}|\).

Notice in the applet that when \(A\) and \(B\) are moved away from each other, the bisector has an increasing segment that is common to the Euclidean bisector (not really surprising given the quotient between the distance from \(A\) to \(B\) in the taxicab geometry and in the euclidean metric, and how this quotient behaves when \(A\) and \(B\) pull away).