Sphere and Line Intersection Calculator

Points of Intersection

To find the points of intersection between a sphere and a line in 3D space, you can use the parametric equations of the line and substitute them into the equation of the sphere.
The general equation of a sphere is

$(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2 \quad (I)$ , and the parametric equations of a line are

$x = x_0 + at \; , \; y = y_0 + bt \; , \; \text{and} \; z = z_0 + ct \quad (II)$ .
The point(s) of intersection are found by solving the above system of equation as follows:
Substitute

$x ,y , z$ in equaion (I) by their expressions in (II) to obtain the equation:

$( x_0 + at - h)^2 + (y_0 + bt - k)^2 + (z_0 + ct - l)^2 = r^2$ Expand the above equation to obtain a quadratic equation in one variable

$t$ , solve it to find

$t$ and substitute into equations (II) to find the point of intersection

$(x,y,z)$

Use of Calculator

Enter the parameters of the sphere

$(h, k, l, r)$ : Enter the parameters of the line

$(x_0, y_0, z_0, a, b, c)$ :

Sphere and Line Intersection Calculator

Points of Intersection

Use of Calculator

Result: