Intersection with FindInstance

Question

How to find individual root parameters pair ( t, T) to find curve intersections using FindInstance? Hopefully it would output two, instead of 6 pairs in the following:

ParametricPlot[{{2 t + 1, 2 - t^2}  , {   t^3 , t^2 - 4}} , {t, -2.3, 
  2}, PlotStyle -> {{Thick, Red}, {Thick, Blue}}, 
 GridLines -> Automatic]
NSolve[{2 t + 1, 2 - T^2}  == {t^3 , T^2 - 4}  , {t, T}]

Like e.g., given by J.M. and Chris in recent:

Points of Intersection

or may be required to use Reduce?

EDIT1:

How do we selectively choose solution sets from among the roots of NSolve?

Answer 1

The problem lies in the location of t and T as variables .

To see this, note that we have two parametric curves:

f1[t_] := {2 t + 1, 2 - t^2};
f2[t_] := {t^3, t^2 - 4};

We know the intersections must occur for some values t and T such that f1[t]==f2[T].

NSolve[f1[t] == f2[T], {t, T}]

or more explicitly (note the different positioning of t and T)

NSolve[{2 t + 1, 2 - t^2} == {T^3, T^2 - 4}, {t, T}]

This gives six solutions (Mathematica output omitted). To get the two real solutions seen in the parametric plot, use the additional argument Reals.

intersect=NSolve[{2 t + 1, 2 - t^2} == {T^3, T^2 - 4}, {t, T},Reals]
(* {{t -> -1.98369, T -> -1.437}, {t -> 1.79866, T -> 1.66278}} *)

These two solutions correspond to the points

f1[t]/.intersect[[1]]
f2[T]/.intersect[[1]]
(* {-2.96738, -1.93502} *)
(* {-2.96738, -1.93502} *)

and

f1[t]/.intersect[[2]]
f2[T]/.intersect[[2]]
(* {4.59731,-1.23516} *)
(* {4.59731,-1.23516} *)

which indeed appear to be the intersections in the parametric plot.

Alternate method using FindInstance

It is also possible to solve this problem using

intersect2 = FindInstance[2 t + 1 == T^3 && 2 - t^2 == T^2 - 4, {t, T}, Reals, 2] // N
(* {{t -> -1.98369, T -> -1.437}, {t -> 1.79866, T -> 1.66278}} *)

This gives the same solution as above, although the NSolve method appears to be slightly faster.

AbsoluteTiming[intersect = NSolve[{2 t + 1, 2 - t^2} == {T^3, T^2 - 4}, {t,T}, Reals]]
AbsoluteTiming[intersect2 = FindInstance[2 t + 1 == T^3 && 2 - t^2 == T^2 - 4, {t, T}, Reals, 2] // N]
(* 0.005154, ... *)
(* 0.016175, ... *)

Answer 2

Edit: Using FindInstance

a = 2 t + 1; b = 2 - t^2; c = t^3; d = t^2 - 4;
eq1 = Eliminate[{x == a, y == b}, t]

7 - 4 y == -2 x + x^2

eq2 = Eliminate[{x == c, y == d}, {t}]

64 + 48 y + 12 y^2 + y^3 == x^2

fi = FindInstance[Evaluate@eq1 && Evaluate@eq2, {x, y}, Reals, 2] // N

{{x -> -2.96738, y -> -1.93502}, {x -> 4.59731, y -> -1.23516}}

ContourPlot[{Evaluate@eq1, Evaluate@eq2}, {x, -10, 10}, {y, -5, 5}, 
 Epilog -> {Red, PointSize[Large], Point[{x, y}] /. fi}]

---

f1[t_] := {2 t + 1, 2 - t^2}

IntersectionRules = NSolve[{2 t + 1, 2 - t^2} == {T^3, T^2 - 4}, {t, T}, Reals]

{*
{t -> -1.98369, T -> -1.437}, {t -> 1.79866, T -> 1.66278}
*}

IntersectionPoints = {f1[t]} /. IntersectionRules

{*
{{-2.96738, -1.93502}}, {{4.59731, -1.23516}}
*}

ParametricPlot[{{2 t + 1, 2 - t^2}, {t^3, t^2 - 4}}, {t, -5, 5}, 
 PlotRange -> 9, Epilog -> {Red, PointSize[Large], Point /@ IntersectionPoints}]