<Text-field layout="Heading 1" style="Heading 1">Outline of Lesson 9</Text-field>

<Text-field bookmark="9.A" layout="Heading 1" style="Heading 1">9.A Trajectories Orthogonal to a Family of Circles</Text-field>

Two curves NiMvJSJ5Ry0lImZHNiMlInhH and NiMvJSJ5Ry0lImdHNiMlInhH that intersect at NiMvJSJ4RyUiYUc= are orthogonal at NiMvJSJ4RyUiYUc= if the tangents to each curve are perpendicular at NiMvJSJ4RyUiYUc=. Thus, the condition

NiMvKiYtJSRmfidHNiMlImFHIiIiLSUjZydHRidGKSwkRikhIiI=

is necessary and sufficient for the two curves to be orthogonal at NiMvJSJ4RyUiYUc=. This condition says nothing more than that the slopes are negative reciprocals of each other at NiMvJSJ4RyUiYUc=.

Two families of curves defined implicitly by NiMvLSUiRkc2JCUieEclInlHJSZhbHBoYUc= and NiMvLSUiR0c2JCUieEclInlHJSViZXRhRw== are orthogonal if any member of one family that intersects a curve of the other family, does so at right angles. To understand what it means for two families of curves to be orthogonal, consider the following example.

The family of circles centered at (NiMlImNH, 0) with radius NiMlImNH can be described as NiMvLCYqJCwmJSJ4RyIiIiUiY0chIiIiIiNGKCokJSJ5R0YrRigqJEYpRis=, or NiMvLSUiRkc2JCUieEclInlHIiIh with

F := unapply( simplify( (x-c)^2+y^2-c^2 ), (x,y) ): 'F'(x,y) = F(x,y);

Representative examples of this family of curves are seen in Figure 9.1.

plotF := C -> implicitplot( eval(F(x,y),c=C), x=-20..20, y=-10..10, scaling=constrained, grid=[60,60] ): p1 := display( seq( plotF(c), c=-9..9) ): display(p1, title="Figure 9.1");

To find the family of curves orthogonal to this family of circles, obtain the derivative for any curve of the family, expressed in a form independent of the parameter NiMlImNH. The derivative of a member of the family can be obtained by implicit differentiation of the equation

NiMvLSUiRkc2JCUieEctJSJ5RzYjRiciIiE=

The result of implicit differentiation is

diff_F := diff( F(x,y(x))=0, x );

and the derivative NiMqJiUjZHlHIiIiJSNkeEchIiI= is

diff_with_c := simplify(isolate(diff_F, diff(y(x),x)));

Unfortunately, this expression for the derivative contains the parameter NiMlImNH. That makes the expression dependent on the particular curve that NiMlImNH selects from the family. The desired expression for the derivative must be independent of this parameter. Hence, eliminate NiMlImNH by use of the equation NiMvLSUiRkc2JCUieEclInlHIiIh. Either solve for NiMlImNH and substitute, or use

q := eliminate({F(x,y(x)),diff_with_c},c);

The desired expression is implicitly contained in

q1 := q[2][1]= 0;

This equation can be solved for NiMqJiUjZHlHIiIiJSNkeEchIiI= and the negative reciprocal obtained, or the derivative NiMqJiUjZHlHIiIiJSNkeEchIiI= can be replace with NiMsJComIiIiRiUtJSN5J0c2IyUieEchIiJGKg== to produce the differential equation of the orthogonal family. The second approach gives the ODE

q2 := eval(q1, diff(y(x),x) = -1/diff(y(x),x));

The solution of this equation is

q3 := dsolve(q2, y(x), implicit);

Combining the logarithms and exponentiating leads to

q4 := simplify(map(exp,combine(q3,ln,anything,symbolic)));

Change the constant of integration to a simpler name to obtain

q5 := eval(q4, exp(-_C1)=2*a);

Either multiply through by the denominator, or bring all terms to the left, add fractions, and select the numerator of the resulting expression on the left. This would be

q6 := -numer(normal(lhs(q5)-rhs(q5))) = 0;

To obtain a graph of members of this new family, use the same technique that produced Figure 9.1. Begin by making the left side into the function NiMtJSJHRzYkJSJ4RyUieUc=.

G := unapply(eval(lhs(q6), y(x)=y),(x,y)): 'G'(x,y) = G(x,y);

Then, Figure 9.2 contains members of the family orthogonal to the family described by the equation NiMvLSUiRkc2JCUieEclInlHIiIh.

plotG := C -> implicitplot( eval(G(x,y),a=C), x=-20..20, y=-20..20, scaling=constrained, grid=[60,60], color=blue ): p2 := display( seq( plotG(a), a=-9..9) ): display(p2, title="Figure 9.2");

Figure 9.3 superimposes the curves of the two families.

display([p1,p2], title="Figure 9.3");