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##### Intersect

This article is about GeoGebra command.

##### Command Categories (All commands)

Intersect( <Object>, <Object> )
Yields the intersection points of two objects.
Examples:
• 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> )
Examples:
• `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:
• to get all the intersection points in a list you can use eg `{Intersect(a,b)}`
• See also IntersectConic and IntersectPath commands.
• See also Intersect tool.
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