# Intersect Command

From GeoGebra Manual

- 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).

- 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> )
**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.