In analytic geometry

Analytic geometry

Analytic geometry, or analytical geometry has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties...

, a line

Line (mathematics)

The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...

 and a sphere

Sphere

A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

 can intersect

Intersection (set theory)

In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....

 in three ways: no intersection at all, at exactly one point, or in two points. Methods for distinguishing these cases, and determining equations for the points in the latter cases, are useful in a number of circumstances. For example, this is a common calculation to perform during ray tracing

Ray tracing (physics)

In physics, ray tracing is a method for calculating the path of waves or particles through a system with regions of varying propagation velocity, absorption characteristics, and reflecting surfaces. Under these circumstances, wavefronts may bend, change direction, or reflect off surfaces,...

 (Eberly 2006:698).

Calculation using vectors in 3D

In vector notation

Vector notation

This page is an overview of the common notations used when working with vectors, which may be spatial or more abstract members of vector spaces....

, the equations are as follows:

Equation for a sphere

Sphere

A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

• - center point
• - radius

Equation for a line starting at (0,0,0)

• - distance along line from starting point
• - direction of line (a unit vector)

Solving for :

Equations combined

Expanded

Rearranged

Quadratic formula and simplified. (This quadratic equation is an example of Joachimsthal's Equation).

Note that is a unit vector, and thus . Thus, we can simplify this further to

• If the value under the square-root () is less than zero, then it is clear that no solutions exist, i.e. the line does not intersect the sphere (case 1).
• If it is zero, then exactly one solution exists, i.e. the line just touches the sphere in one point (case 2).
• If it is greater than zero, two solutions exist, and thus the line touches the sphere in two points (case 3).

See also

• Analytic geometry
  Analytic geometry
  Analytic geometry, or analytical geometry has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties...
  
  
• Line-plane intersection
  Line-plane intersection
  In analytic geometry, the intersection of a line and a plane can be the empty set,a point, ora line. Distinguishing these cases, and determining equations for the point and line in the latter cases have use, for example, in computer graphics, motion planning, and collision detection.-Parametric...
  
  
• Line of intersection between two planes
  Plane (mathematics)
  In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...