the hyperbolas of Apollonius

After setting up all of the above routines, we then went to the history books and discovered the work of Apollonius.

Apollonius did not have the tools of calculus or analytic geometry to explore this geometry, so he used ordinary geometry instead, which made his discussions very complicated. He studied many properties of the conic sections and in particular investigated the distance problem for each of them, namely how to construct the shortest and longest connecting line segments to the conic section from a given point in the plane, and how the distance between the given point and a point on the ellipse varied as one moved away from these special points on the ellipse where the distance is extremal. A construction with similar triangles leads to an equation of a hyperbola whose intersections with the ellipse are the points from which normals pass through the given point. Our construction reobtains his results analytically but not following his geometric approach, which is not easily digested. This hyperbola is a great help in visualizing what happens to the normals as one moves the given point around in the plane.

Here is the hyperbola of Apollonius incorporated into the previous procedure to show the resulting geometry.

the extended procedure ellipser_hyp(b,r,theta)

>    ellipser_hyp(.75,0.2,-Pi/4);

[Maple Plot]

For this particular configuration, the major axis is along the x -axis, which is important in this derivation.


The rightmost point P on the ellipse in the first quadrant from which a normal passes through the given point O in the fourth quadrant is a configuration for which the similar triangle construction diagram is particularly simple. Dropping a perpendicular to the
x -axis from O defines the point M, while doing the same from P defines the point N, while let G be the point where the normal line crosses the x -axis, and finally let T be the intersection of the tangent line to the ellipse at P with the x -axis.

setting up the diagram

>    #plotsetup(ps,plotoptions=`noborder,portrait`,plotoutput=`c:/local/fig_triangle_old.eps`);

>    Triangle_Figure;

>    #plotsetup(inline):

[Maple Plot]

Letting s  and t  be the lengths of MG and GN, it follows that

>    eq0:=s+t = x-x0;

eq0 := s+t = x-x0

Triangles GMO, GNP and PNT are all similar right triangles, leading to two equalities:

>    eq1:= s/(-y0)=t/y;
eq2:=t/y =y/(a^2/x-x);

eq1 := -s/y0 = t/y

eq2 := t/y = y/(a^2/x-x)

where a^2/x  is the x -intercept of the tangent line ( x -coordinate of the point T). This follows from using calculus to obtain the equation of the tangent line to the ellipse at P, which can be written using ( u, v ) as the coordinates since ( x, y ) are the coordinates of the point of tangency, from which setting v = 0  and solving for u  yields the x -intercept:

>    u*x/a^2+v*y/b^2=1;
solve(subs(v=0,%),u);

u*x/a^2+v*y/b^2 = 1

a^2/x

From these three equations, one can eliminate s  and t , leaving only one equation left:

>    s=solve(eq1,s);
subs(%,eq0);
t=solve(%,t);
subs(%,s=solve(eq1,s));
subs(%%,eq2);

s = -t/y*y0

-t/y*y0+t = x-x0

t = (x-x0)*y/(y-y0)

s = -(x-x0)/(y-y0)*y0

(x-x0)/(y-y0) = y/(a^2/x-x)

This last equation just equates the negative reciprocal of the normal line slope to the tangent line slope (the orthogonality condition), so we could have avoided using the similar triangle relationships to get this. Finally, one can simplify this last equation using the equation of the ellipse:

>    eq_ellipse;
expand(%*a^2);
lhs(%)-x^2=rhs(%)-x^2;

x^2/a^2+y^2/b^2 = 1

x^2+a^2*y^2/b^2 = a^2

a^2*y^2/b^2 = a^2-x^2

>    (-x+x0)/(-y+y0) = x*y/(a^2-x^2);
(-x+x0)/(-y+y0) = x*y/(a^2*y^2/b^2);

(-x+x0)/(-y+y0) = x*y/(a^2-x^2)

(-x+x0)/(-y+y0) = x/y/a^2*b^2

>    normal(lhs(%)-rhs(%))=0;

(y*a^2*x-y*a^2*x0-x*b^2*y+x*b^2*y0)/(y-y0)/y/a^2 = 0

>    numer(lhs(%))=0;

y*a^2*x-y*a^2*x0-x*b^2*y+x*b^2*y0 = 0

>    eq_hyperbola:=(b^2-a^2)*x*y-b^2*y0*x+a^2*x0*y=0;

eq_hyperbola := (b^2-a^2)*x*y-x*b^2*y0+y*a^2*x0 = 0

This is the equation of a hyperbola whose asymptotes are a pair of horizontal and vertical lines (since there are no squared terms in the equation) and which passes through both the origin and the given point ( x0, y0 ). By considering translations of the hyperbola x*y = A  to (x-x1)*(y-y1) = A , and requiring that A = x1*y1  to eliminate the constant term shows that the center of the hyperbola has coordinates:

>    [x1,y1]=[a^2*x0/(a^2-b^2),-b^2*y0/(a^2-b^2)];

[x1, y1] = [a^2*x0/(a^2-b^2), -b^2*y0/(a^2-b^2)]

Heath (p.166) refers only to similar triangles GMO and GNP and the corresponding eq1 of similar sides in re-interpreting the work Apollonius using analytic geometry. Somehow Apollonius obtained the result that the normals which pass through O come from the points P on the ellipse which are the intersections with this hyperbola, although he did not write down any equations for it.

Note that the x -coordinate of the point G, which is the intercept of the normal line, is just:

>    'xG'=x0+s;
subs(s = -(-x+x0)/(-y+y0)*y0,'%');

xG = x0+s

xG = x0-(-x+x0)/(-y+y0)*y0

which we used in placing the letter G in the above graphic.

To recap, we needed only 3 inputs to derive the equation of the hyperbola of Apollonius:  the equation of the ellipse itself, the intercept a^2/x   with its major axis of the tangent line from the point ( x , y ) on the ellipse, and the orthogonality condition between the tangent line segment and the normal line from the given point.

>    eq_ellipse;
'innerprod([x-x0,y-y0],[a^2/x-x,0-y])'=0;

x^2/a^2+y^2/b^2 = 1

innerprod([x-x0, y-y0],[a^2/x-x, -y]) = 0

intersection of the ellipse and hyperbola:
nonparametric approach (hindsight)

>   

properties of the hyperbola

the extended procedure ellipser_hyp_center(b,r,theta) and its animation

With the animation one sees how the situation develops as the given point (x0,y0) = ( r*cos(theta), r*sin(theta) ) moves out on a ray from the origin at a fixed angle. The center of the hyperbola moves out on the other side of the (major) x -axis at a smaller angle, while the vertices of the hyperbola move out from its center, all at distances proportional to r . Inside the evolute the branch of the hyperbola not passing through the given point moves out from the center of the ellipse still intersecting the ellipse in two points, but as the given point passes through the evolute, the second branch becomes tangent to the ellipse and then moves out of the ellipse, reducing the number of normals to 3 on the evolute and 2 outside the evolute, provided one is not moving along one of the axes of the ellipse. In that case the vertices of the ellipse on that axis remain as two intersection points, while the remaining pair degenerates to one of these semi-axis vertices as the given point reaches the cusp of the evolute, so 3 become 1, the total dropping from 4 to 2 directly.