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Medial Curves

Point-Point

Point-Line

Point-Circle

Line-Line

Line-Circle

Circle-Circle

Explorations

A medial curve is the locus of points equidistant from two loci of points. In this chapter we will derive the equations of medial curves that are equidistant from two points, a point and a curve (line or circle) and two curves (lines or circles).

Initialize

To initialize Descarta2D, select the input cell bracket and press SHIFT-Enter.

This initialization assumes that the Descarta2D software has been copied into one of the standard directories for AddOns which are on the Mathematica search path, $Path.

<<Descarta2D`

Point-Point [Top]

Consider two distinct points and and a point P(x,y). The distance, , from P to is given by

Likewise, the distance, , from P to is given by

If point P is on the medial curve defined by and , then and

Squaring both sides of this equation and rearranging yields

which is easily recognized in this form as the general equation of a line. The medial line is the perpendicular bisector of the line segment joining and . The derivation is provided in the exploration mdptpt.html.

Example: Find the equation of the medial line determined by the two points (1,2) and (-1,-1). Plot the points and the medial line.

Solution: The function MedialLoci2D[{point,point}] returns a list of one line that is the medial line determined by the two points.

l12=MedialLoci2D[{p1=Point2D[{1,2}],

p2=Point2D[{-1,-1}]}]

Sketch2D[{p1,p2,l12}]

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Descarta2D Hint: The function Point2D[point,point,Perpendicular2D] returns the perpendicular bisector of the line segment joining two points. This function may also be used.

Point-Line [Top]

Consider the point and the line , where (to simplify the derivation, the coefficients of the line are normalized because distance is involved). The distance, , from a point P(x,y) to is given by

The distance, , from a point P(x,y) to the normalized line is given by

Since P is the locus of points on the medial curve, , and by squaring and rearranging we obtain the quadratic equation

where

A | = | |

B | = | |

C | = | |

D | = | |

E | = | |

F | = |

These equations are derived in the exploration mdptln.html.

The definition of a parabola is the locus of points equidistant from a point and a line, so it is obvious that in the general case the medial curve of a point and a line will be a parabola.

Example: Find the medial curve of the point (-1,-1) and -x-y+1=0 and plot.

Solution: The function MedialLoci2D[{point,line}] returns a list of one curve that is the medial curve of the point and the line.

crv1=MedialLoci2D[{p1=Point2D[{-1,-1}],

l2=Line2D[-1,-1,1]}]

Sketch2D[{p1,l2,crv1}]

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If the point is on line , then the medial curve will be a line perpendicular to the defining line.

Example: Find the medial curve of the point (1,0) and the line -x-y+1=0 and plot. Notice that the point is on the line.

Solution: The same function, MedialLoci2D[{point,line}], introduced in the previous example will return a list containing the medial curve, which is a line in this case.

crv1=MedialLoci2D[{p1=Point2D[{1,0}],

l2=Line2D[-1,-1,1]}]

Sketch2D[{p1,l2,crv1}]

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Point-Circle [Top]

Consider a point and a circle with center and radius . The distance, , from a point P(x,y) to is given by

The distance, , from a point P(x,y) to the circle is given by

when P is outside of circle . When P is inside the distance, , is given by

If P is the locus of points equidistant from and , then . Squaring both sides of this equation eliminates the distinction between points P inside the circle and outside the circle. Rearranging the resulting equation yields the quadratic equation

where

A | = | |

B | = | |

C | = | |

D | = | |

E | = | |

F | = | |

R | = |

This derivation is included in the exploration mdptcir.html.

If the point is outside circle , the medial curve will be a hyperbola. If is inside , the medial curve will be an ellipse. In the special case that is on , the medial curve will be a line containing the center point of . If is coincident with the center of , then the medial curve will be a circle centered at with a radius of .

Example: Find the medial curves of four points (-8,1), (-4,1), (-2,1) and (0,1) with the circle . Plot each of the curves separately.

Solution: The Descarta2D function MedialLoci2D[{point,circle}] returns a list of one object equidistant from the point and the circle.

pts={Point2D[{-8,1}],Point2D[{-4,1}],

Point2D[{-2,1}],Point2D[{0,1}]};

c2=Circle2D[{0,1},4];

crvs=Map[MedialLoci2D[{#,c2}]&,pts]

Map[Print[Sketch2D[{pts[[#]],c2,crvs[[#]]}]]&, {1,2,3,4}];

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Line-Line [Top]

The locus of points equidistant from two lines

. | and | |

. | . |

are the two angle bisector lines. The equations of these two lines are

as shown in the exploration mdlnln.html.

Example: Find the medial lines for 3x-4y+1=0 and 2x+2y-3=0 and plot.

Solution: The function MedialLoci2D[{line,line}] returns a list of lines that are the medial lines of the two given lines. If the lines are parallel, then the list will contain one line; otherwise, it will contain two lines.

lns=MedialLoci2D[{l1=Line2D[3,-4,1],

l2=Line2D[2,2,-3]}]

Sketch2D[{l1,l2,lns}]

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Line-Circle [Top]

Consider a line , where (to simplify the derivation, the coefficients of the line are normalized because distance is involved), and a circle with center at and radius . The distance, , from a point P(x,y) to line is given by

The distance, , from point P(x,y) to circle is given by

when P is outside of circle . When P is inside the distance, , is given by

We introduce a sign constant, s, which takes on the values ±1, so that we can combine the two equations for yielding

If P is the locus of points equidistant from and , then . Rearranging the resulting equation yields the quadratic equation

where

A | = | |

B | = | |

C | = | |

D | = | |

E | = | |

F | = |

This derivation is included in mdlncir.html. If the line intersects the circle in two distinct points, then the medial curves will be two parabolas, each passing through the points of intersection of the line and the circle.

Example: Find the curves that are equidistant from the line y=1 and the circle . Plot the curves.

Solution: The function MedialLoci2D[{line,circle}] returns a list of curves equidistant from a line and a circle.

l1=Line2D[0,1,-1];

c2=Circle2D[{0,1},2];

crvs=MedialLoci2D[{l1,c2}]

Sketch2D[{l1,c2,crvs}]

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If the line is tangent to the circle then one of the medial curves will be a parabola, and the other will be a line passing through the tangency point and the center point of the circle. Strictly speaking, not all of the points on the line are equidistant from the line and the circle, unless we consider the distance to be measured both from the closest point on the circle and the farthest point on the circle.

Example: Find the curves that are equidistant from the line y=3 and the circle and plot. Notice that the line is tangent to the circle.

Solution: Use the function MedialLoci2D[{line,circle}] introduced in the previous example.

l1=Line2D[0,1,-3];

c2=Circle2D[{0,1},2];

crvs=MedialLoci2D[{l1,c2}]

Sketch2D[{l1,c2,crvs}]

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If the line and the circle do not intersect, then the two medial curves will be parabolas. Strictly speaking, only one of these parabolas is equidistant from the circle and the line, unless we consider the distance to be measured both from the closest point on the circle and the farthest point on the circle.

Example: Find the curves that are equidistant from the line y=5 and the circle . Plot the curves.

Solution: Use the function MedialLoci2D[{line,circle}] as described in the previous examples.

l1=Line2D[0,1,-5];

c2=Circle2D[{0,1},2];

crvs=MedialLoci2D[{l1,c2}]

Sketch2D[{l1,c2,crvs}]

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Circle-Circle [Top]

Consider two distinct circles

and .

Using the same distance equating techniques outlined in previous sections, and introducing a sign constant s=±1, we can obtain the quadratic equation of the curves equidistant from the two circles

where

A | = | |

B | = | |

C | = | |

D | = | |

E | = | |

F | = |

and

R | = | |

= | ||

= | ||

s | = | ±1. |

This derivation is included in the exploration mdcircir.html. Table [medial:tbl01] summarizes the medial curves associated with a pair of circles in several configurations taking into consideration differing radii and equal radii. Strictly speaking, some of the branches of these curves are not equidistant from the two circles, unless we consider the distance to be measured both from the closest and the farthest point on the circles.

Medial curves for two circles.

Example: Find and plot the curves equidistant from the two circles and .

Solution: Use the function MedialLoci2D[{circle,circle}].

c1=Circle2D[{0,0},3];

c2=Circle2D[{0,2},2];

crvs=MedialLoci2D[{c1,c2}]

Sketch2D[{c1,c2,crvs}]

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Descarta2D Hint: The function MedialLoci2D[{obj,obj}] produces the same result as MedialLoci2D[{obj,obj}], that is, the objects may be provided in any order in the list. In addition, MedialEquations2D[{obj,obj}] will return a list of lines and/or quadratics representing the medial curves.

Explorations [Top]

Medial Curve, Point-Point. [mdptpt.html]

Show that the line is equidistant from the points and .

Medial Curve, Point-Line. [mdptln.html]

Show that the quadratic equation

where

A | = | |

B | = | |

C | = | |

D | = | |

E | = | |

F | = |

is equidistant from the point and the line , assuming that L is normalized ().

Medial Curve, Point-Circle. [mdptcir.html]

Show that the quadratic equation

where

A | = | |

B | = | |

C | = | |

D | = | |

E | = | |

F | = | |

R | = |

is equidistant from the point and the circle

Medial Curve, Line-Line. [mdlnln.html]

Show that the pair of lines whose equations are

is equidistant from the two lines and .

Medial Curve, Line-Circle. [mdlncir.html]

Show that the two quadratics whose equations are given by

where

A | = | |

B | = | |

C | = | |

D | = | |

E | = | |

F | = | |

s | = | ±1 |

are equidistant from the line

and the circle

assuming .

Medial Curve, Circle-Circle. [mdcircir.html]

Show that the two quadratics whose equations are given by

where

A | = | |

B | = | |

C | = | |

D | = | |

E | = | |

F | = |

and

R | = | |

= | ||

= | ||

s | = | ±1 |

are equidistant from the two circles

Medial Curve Type. [mdtype.html]

Show that the medial curve equidistant from a point and a circle is a hyperbola when the point is outside the circle and it is an ellipse when the point is inside the circle. (Hint: Examine the value of the discriminant of the medial quadratic.)

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