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Medial Curves

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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).

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<<Descarta2D`

Point-Point [Top]

Consider two distinct points "medial_1.gif" and "medial_2.gif" and a point P(x,y). The distance, "medial_3.gif", from P to "medial_4.gif" is given by

"medial_5.gif"

Likewise, the distance, "medial_6.gif", from P to "medial_7.gif" is given by

"medial_8.gif"

If point P is on the medial curve defined by "medial_9.gif" and "medial_10.gif", then "medial_11.gif" and

"medial_12.gif"

Squaring both sides of this equation and rearranging yields

"medial_13.gif"

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 "medial_14.gif" and "medial_15.gif". 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}]}]

"medial_16.gif"

Sketch2D[{p1,p2,l12}]

"medial_17.gif"

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 "medial_18.gif" and the line "medial_19.gif", where "medial_20.gif" (to simplify the derivation, the coefficients of the line are normalized because distance is involved). The distance, "medial_21.gif", from a point P(x,y) to "medial_22.gif" is given by

"medial_23.gif"

The distance, "medial_24.gif", from a point P(x,y) to the normalized line "medial_25.gif" is given by

"medial_26.gif"

Since P is the locus of points on the medial curve, "medial_27.gif", and by squaring and rearranging we obtain the quadratic equation

"medial_28.gif"

where

A = "medial_29.gif"
B = "medial_30.gif"
C = "medial_31.gif"
D = "medial_32.gif"
E = "medial_33.gif"
F = "medial_34.gif"

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]}]

"medial_35.gif"

Sketch2D[{p1,l2,crv1}]

"medial_36.gif"

If the point "medial_37.gif" is on line "medial_38.gif", 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]}]

"medial_39.gif"

Sketch2D[{p1,l2,crv1}]

"medial_40.gif"

Point-Circle [Top]

Consider a point "medial_41.gif" and a circle "medial_42.gif" with center "medial_43.gif" and radius "medial_44.gif". The distance, "medial_45.gif", from a point P(x,y) to "medial_46.gif" is given by

"medial_47.gif"

The distance, "medial_48.gif", from a point P(x,y) to the circle "medial_49.gif" is given by

"medial_50.gif"

when P is outside of circle "medial_51.gif". When P is inside "medial_52.gif" the distance, "medial_53.gif", is given by

"medial_54.gif"

If P is the locus of points equidistant from "medial_55.gif" and "medial_56.gif", then "medial_57.gif". 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

"medial_58.gif"

where

A = "medial_59.gif"
B = "medial_60.gif"
C = "medial_61.gif"
D = "medial_62.gif"
E = "medial_63.gif"
F = "medial_64.gif"
R = "medial_65.gif"

This derivation is included in the exploration mdptcir.html.

If the point "medial_66.gif" is outside circle "medial_67.gif", the medial curve will be a hyperbola. If "medial_68.gif" is inside "medial_69.gif", the medial curve will be an ellipse. In the special case that "medial_70.gif" is on "medial_71.gif", the medial curve will be a line containing the center point of "medial_72.gif". If "medial_73.gif" is coincident with the center of "medial_74.gif", then the medial curve will be a circle centered at "medial_75.gif" with a radius of "medial_76.gif".

Example: Find the medial curves of four points (-8,1), (-4,1), (-2,1) and (0,1) with the circle "medial_77.gif". 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]

"medial_78.gif"

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

"medial_79.gif"

"medial_80.gif"

"medial_81.gif"

"medial_82.gif"

Line-Line [Top]

The locus of points equidistant from two lines

. "medial_83.gif" and
. "medial_84.gif" .

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

"medial_85.gif"

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]}]

"medial_86.gif"

Sketch2D[{l1,l2,lns}]

"medial_87.gif"

Line-Circle [Top]

Consider a line "medial_88.gif", where "medial_89.gif" (to simplify the derivation, the coefficients of the line are normalized because distance is involved), and a circle "medial_90.gif" with center at "medial_91.gif" and radius "medial_92.gif". The distance, "medial_93.gif", from a point P(x,y) to line "medial_94.gif" is given by

"medial_95.gif"

The distance, "medial_96.gif", from point P(x,y) to circle "medial_97.gif" is given by

"medial_98.gif"

when P is outside of circle "medial_99.gif". When P is inside "medial_100.gif" the distance, "medial_101.gif", is given by

"medial_102.gif"

We introduce a sign constant, s, which takes on the values ±1, so that we can combine the two equations for "medial_103.gif" yielding

"medial_104.gif"

If P is the locus of points equidistant from "medial_105.gif" and "medial_106.gif", then "medial_107.gif". Rearranging the resulting equation yields the quadratic equation

"medial_108.gif"

where

A = "medial_109.gif"
B = "medial_110.gif"
C = "medial_111.gif"
D = "medial_112.gif"
E = "medial_113.gif"
F = "medial_114.gif"

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 "medial_115.gif". 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}]

"medial_116.gif"

Sketch2D[{l1,c2,crvs}]

"medial_117.gif"

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 "medial_118.gif" 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}]

"medial_119.gif"

Sketch2D[{l1,c2,crvs}]

"medial_120.gif"

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 "medial_121.gif". 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}]

"medial_122.gif"

Sketch2D[{l1,c2,crvs}]

"medial_123.gif"

Circle-Circle [Top]

Consider two distinct circles

"medial_124.gif" and "medial_125.gif".

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

"medial_126.gif"

where

A = "medial_127.gif"
B = "medial_128.gif"
C = "medial_129.gif"
D = "medial_130.gif"
E = "medial_131.gif"
F = "medial_132.gif"

and

R = "medial_133.gif"
"medial_134.gif" = "medial_135.gif"
"medial_136.gif" = "medial_137.gif"
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_138.gif"

Medial curves for two circles.

Example: Find and plot the curves equidistant from the two circles "medial_139.gif" and "medial_140.gif".

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

c1=Circle2D[{0,0},3];
c2=Circle2D[{0,2},2];
crvs=MedialLoci2D[{c1,c2}]

"medial_141.gif"

Sketch2D[{c1,c2,crvs}]

"medial_142.gif"

Descarta2D Hint: The function MedialLoci2D[{obj"medial_143.gif",obj"medial_144.gif"}] produces the same result as MedialLoci2D[{obj"medial_145.gif",obj"medial_146.gif"}], that is, the objects may be provided in any order in the list. In addition, MedialEquations2D[{obj"medial_147.gif",obj"medial_148.gif"}] 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 "medial_149.gif" is equidistant from the points "medial_150.gif" and "medial_151.gif".

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

Show that the quadratic equation

"medial_152.gif"

where

A = "medial_153.gif"
B = "medial_154.gif"
C = "medial_155.gif"
D = "medial_156.gif"
E = "medial_157.gif"
F = "medial_158.gif"

is equidistant from the point "medial_159.gif" and the line "medial_160.gif", assuming that L is normalized ("medial_161.gif").

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

Show that the quadratic equation

"medial_162.gif"

where

A = "medial_163.gif"
B = "medial_164.gif"
C = "medial_165.gif"
D = "medial_166.gif"
E = "medial_167.gif"
F = "medial_168.gif"
R = "medial_169.gif"

is equidistant from the point "medial_170.gif" and the circle

"medial_171.gif"

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

Show that the pair of lines whose equations are

"medial_172.gif"

is equidistant from the two lines "medial_173.gif" and "medial_174.gif".

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

Show that the two quadratics whose equations are given by

"medial_175.gif"

where

A = "medial_176.gif"
B = "medial_177.gif"
C = "medial_178.gif"
D = "medial_179.gif"
E = "medial_180.gif"
F = "medial_181.gif"
s = ±1

are equidistant from the line

"medial_182.gif"

and the circle

"medial_183.gif"

assuming "medial_184.gif".

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

Show that the two quadratics whose equations are given by

"medial_185.gif"

where

A = "medial_186.gif"
B = "medial_187.gif"
C = "medial_188.gif"
D = "medial_189.gif"
E = "medial_190.gif"
F = "medial_191.gif"

and

R = "medial_192.gif"
"medial_193.gif" = "medial_194.gif"
"medial_195.gif" = "medial_196.gif"
s = ±1

are equidistant from the two circles

"medial_197.gif"

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 "medial_198.gif" of the medial quadratic.)


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