Intersect Command

Intersect( <Object>, <Object> )

Yields the intersection points of two objects.

  • 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.

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.

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.

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.

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.

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> )

  • 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, …​)