# Intersect Command

From GeoGebra Manual

- 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).

- Let

- Intersect[ <Object>, <Object>, <Index of Intersection Point> ]
- Yields the n
^{th}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:**

- 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.