Intersect Command

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Intersect[ <Object>, <Object> ]
Yields the intersection points of two objects.
Example:
  • Let a: -3x + 7y = -10 be a line and c: x^2 + 2y^2 = 8 be an ellipse. Intersect[a, c] yields the intersection points E = (-1.02, -1,87) and F = (2.81, -0.22) of the line and the ellipse.
  • Intersect[y = x + 3, Curve[t, 2t, t, 0, 10]] yields A=(3,6).
  • Intersect[Curve[2s, 5s, s,-10, 10 ], Curve[t, 2t, t, -10, 10]] yields A=(0,0).
Intersect[ <Object>, <Object>, <Index of Intersection Point> ]
Yields the nth intersection point of two objects. Each object must be a line, conic, polynomial function or implicit curve.
Example:
Let a(x) = x^3 + x^2 - x be a function and b: -3x + 5y = 4 be a line. Intersect[a, b, 2] yields the intersection point C = (-0.43, 0.54) of the function and the line.
Intersect[ <Object>, <Object>, <Initial Point> ]
Yields an intersection point of two objects by using a numerical, iterative method with initial point.
Example:
Let a(x) = x^3 + x^2 - x be a function, b: -3x + 5y = 4 be a line, and C = (0, 0.8) be the initial point. Intersect[a, b, C] yields the intersection point D = (-0.43, 0.54) of the function and the line by using a numerical, iterative method.
Intersect[ <Function>, <Function>, <Start x-Value>, <End x-Value> ]
Yields the intersection points numerically for the two functions in the given interval.
Example:
Let f(x) = x^3 + x^2 - x and g(x) = 4 / 5 + 3 / 5 x be two functions. Intersect[ f, g, -1, 2 ] yields the intersection points A = (-0.43, 0.54) and B = (1.1, 1.46) of the two functions in the interval [ -1, 2 ].
Intersect[ <Curve 1>, <Curve 2>, <Parameter 1>, <Parameter 2> ]
Finds one intersection point using a numerical, iterative method starting at the given parameters.
Example:
Let a = Curve[cos(t), sin(t), t, 0, π] and b = Curve[cos(t) + 1, sin(t), t, 0, π].
Intersect[a, b, 0, 2] yields the intersection point A = (0.5, 0.87).


CAS Syntax

Intersect[ <Function>, <Function> ]
Yields a list containing the intersection points of two objects.
Example:
Let f(x):= x^3 + x^2 - x and g(x):= x be two functions. Intersect[ f(x), g(x) ] yields the intersection points list: {(1, 1), (0, 0), (-2, -2)} of the two functions.


Intersect[ <Object>, <Object> ]
Example:
  • Intersect[ <Line> , <Object> ] creates the intersection point(s) of a line and a plane, segment, polygon, conic, etc.
  • Intersect[ <Plane> , <Object> ] creates the intersection point(s) of a plane and segment, polygon, conic, etc.
  • Intersect[ <Conic>, <Conic> ] creates the intersection point(s) of two conics
  • Intersect[ <Plane>, <Plane> ] creates the intersection line of two planes
  • Intersect[ <Plane>, <Polyhedron> ] creates the polygon(s) intersection of a plane and a polyhedron.
  • Intersect[ <Sphere>, <Sphere> ] creates the circle intersection of two spheres
  • Intersect[ <Plane>, <Quadric> ] creates the conic intersection of the plane and the quadric (sphere, cone, cylinder, ...)
Notes:
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