Intersect Command

Intersect[ <Object>, <Object> ]

Yields the intersection points of two objects.

Example:

• Let a: -3x + 7y = -10 be a line with A = (1, -1) and B = (8, 2) and c: x^2 + 2y^2 = 8 be an ellipse with focuses C = (-2, 0) und D = (2, 0). 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[ <Object>, <Object>, <Index of Intersection Point> ]

Yields the n^th intersection point of two objects.

Example:

Let a(x) = x^3 + x^2 - x be a function and b: -3x + 5y = 4 be a line with A = (-3, -1) and B = (2, 2). 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 with A = (-3, -1) and B = (2, 2) 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 for the intervall [ -1, 2 ] the intersection points A = (-0.43, 0.54) and B = (1.1, 1.46) of the two functions.

Intersect[ <Curve 1>, <Curve 2>, <Parameter 1>, <Parameter 2> ]

Finds one intersection point using an 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 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 {(1, 1), (0, 0), (-2, -2)} of the two functions.

Note: See also Intersect Two Objects tool.