How do you know if a line intersects a sphere?
Intersection of a Line and a Sphere (or circle)
- If this is less than 0 then the line does not intersect the sphere.
- If it equals 0 then the line is a tangent to the sphere intersecting it at one point, namely at u = -b/2a.
- If it is greater then 0 the line intersects the sphere at two points.
How do you find the intersection of a plane and a sphere?
To find the intersection with the x z-plane, we set y = 0: (x − 2)+ (z – 4)= -11. Since no points satisfy this equation, the sphere does not intersect the xz-plane. (Also note that the distance from the center of the sphere to the xz-plane is greater than the radius of the sphere.)
Why do great circles intersect at one point?
In the world of spherical geometry, two parallel lines on great circles intersect twice, the sum of the three angles of a triangle on the sphere’s surface exceed 180° due to positive curvature, and the shortest route to get from one point to another is not a straight line on a map but a line that follows the minor arc …
At what point did the two rays intersect?
When two lines, rays, or line segments intersect, they have one common point; in this case, the line segments intersect since they meet at the center of the windmill’s blades. In the figure below, point (3,4) is the intersection of line x = 3 and line y = 4 since that is where the two lines cross.
Is there an intersection between a ray and a sphere?
Note that they can only be an intersection between the ray and the sphere if tca is positive (if it is negative, it means that the vector L and the vector D points in opposite directions. If there is an intersection, it could potentially be behind the ray’s origin but anything that happens behind the ray’s origin is of no use to us).
How to test a ray against a sphere?
We can simply rewrite equation 2 as: where C is the location of the center of the sphere in 3D space. Equation 4 can now be re-written as: | O + tD − C | 2 − R2 = 0. In a more intuitive form, this comes back to say that we can translate the ray by -C and test this transformed ray against the sphere as if it was centered at the origin.
When is there no intersection between a ray and a PC?
If the distance from pc to the ray is greater than the ray then there is no intersection (sphere A in the above figure). If the distance from pc to the ray is equal to the radius of the sphere, then the intersection is a single point: pc (sphere B). The former case requires a little bit extra work.
How to find the ray / line intersection point?
Substitute in the ray equation (1) to find the ray/line intersection point: r*r = dot(C+t*D,C+t*D) = dot(C,C) + 2*dot(C,t*D) + dot(t*D,t*D) or using dot product linearity again: r*r = dot(C+t*D,C+t*D) = dot(C,C) + 2*t*dot(C,D) + t*t*dot(D,D) This is a quadratic equation in t, with constants c=dot(C,C)-r*r, b=2*dot(C,D), and a=dot(D,D).