Chapter 10. Singular Points of Curves

Rovenski Vladimir, Haifa

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10.1 Singular Points of Parametrized Curves

Examples .

1. Let . Then (0,0) is a cuspidal point of the first kind.

> plot([t^2, t^3, t=-1..1], scaling=constrained);

2. Let . Then (0,0) is a cuspidal point of the 2d kind.

> plot([t^2, t^4+t^5, t=-1.2..0.8]);

4. Let us plot the curve and semi-tangent lines to it at singular points.

is a singular point.

> r:=array([2*t-t^2, 3*t-t^3]): rt:=map(diff,r,t);

> p:=plot([r[1], r[2], t=-1.9..2.2], thickness=2): %;

> t1:=solve({rt[1]=0, rt[2]=0}); pp:=subs(t1, [r[1], r[2]]); k:=limit(rt[2]/rt[1], t1); Tp:=pp+[t, k*t];

> q:=plot([Tp[1], Tp[2], t=-2.3..0], linestyle=2):

> plots[display]([p, q], scaling=constrained);

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10.2 Singular Points of Implicitly Defined Plane Curves

Exercise s.

1. The curve (a, b > 0) has the singular point (0,0) of order 3:

Three branches of the curve are tangent to the straight line y=0.

> f:=(a,b)->x^6-2*a*2*x^3*y-b^3*y^3:

> plots[implicitplot](f(1, 1), x=-.5..0.5, y=-.5..0.5, grid=[70,70], axes=framed, scaling=constrained);

3. Relations between a and b when the curve has a double point: .

> F:=y^2-x^3-a*x-b: Fx:=diff(F, x): Fy:=diff(F, y):

> s:=solve({F, Fx, Fy}, {y, a, b});

> FF:=subs({a=-3*t^2, b=2*t^3}, F): FF;

> Y:=t -> solve(FF, y): Y(t);

> plot([seq(Y(t), t={0,2,4})], x=-9..9, y=-17..17);

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10.3 Unusual Singular Points of Plane Curves

Example 1 .

1. The right-hand derivative of the function has a discontinuity ( stopping point)

a t the point x=0.

> p1:=plot(exp(1)^(1/x), x=-5..0, thickness=2): p2:=plot(exp(1)^(1/x), x=.5..5, thickness=2): h:=plot(1, x=-5..5, linestyle=2):

> plots[display]([p1,p2,h], scaling=constrained);

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2. The left-hand derivative of the function has a discontinuity (the angular point) at the point x=0.

> p:=plot(x/(1+exp(1)^(1/x)), x=-1..2, thickness=2): h:=plot(x,x=-1..0, linestyle=2): g:=plot(0,x=0..2, linestyle=2):

> plots[display]([p,h,g], scaling=constrained);

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