According to the number of real solutions of t, we could determine if the ray intersection occurs. Here, if no real roots, then no intersection; only one real root means that the ray intersects with the sphere at edge; and two real roots means that there two points intersecting with the sphere.

nikyarora

Since r is normally normalized, we can say that a = 1 generally because the result of the dot product of a normalized vector with itself is 1. However, when these rays don't have their direction normalized, we cannot make this generalization.

briana-jin-zhang

The silhouette of an object is when there is exactly one intersection of the ray and object (sphere in this case). Interesting fact: the projection of a sphere onto an image plane in the pinhole camera model will always yield an ellipse.

briana-jin-zhang

Actually I misspoke, it can also appear as a parabola or hyperbola since the rays intersection with the sphere only once will form a cone, and the intersection of planes and cones yield all conic sections.

According to the number of real solutions of t, we could determine if the ray intersection occurs. Here, if no real roots, then no intersection; only one real root means that the ray intersects with the sphere at edge; and two real roots means that there two points intersecting with the sphere.

Since r is normally normalized, we can say that a = 1 generally because the result of the dot product of a normalized vector with itself is 1. However, when these rays don't have their direction normalized, we cannot make this generalization.

The silhouette of an object is when there is exactly one intersection of the ray and object (sphere in this case). Interesting fact: the projection of a sphere onto an image plane in the pinhole camera model will always yield an ellipse.

Actually I misspoke, it can also appear as a parabola or hyperbola since the rays intersection with the sphere only once will form a cone, and the intersection of planes and cones yield all conic sections.