The curvature sphere provides information of the turning rate at a point on
the curve. The center of the curvature sphere is always on the positive
direction of the normal vector. The ** curvature** at a
point on the curve is the reciprocal of the radius of the curvature sphere.
As a result, the larger the curvature sphere, the smaller the curvature
and the flatter the curve. If a curve has a very sharp turn at a point, the
curvature at that point must be very large and the curvature sphere at that
point will be very small. On the other hand, if a a curve is flat at a
point, the curvature is very small and the curvature sphere will be very
larger. Consequently, a straight line has curvature zero because there is no
turn on a line, and the curvature sphere will have radius an infinite
value at any point on a line. Based on this discussion, it is clear that
examining curvature spheres will help you understand if a curve has
sharp turns.

To display curvature sphere, you must first activate
**Tangent, Binormal & Normal**.
Once the triad is displayed, click on
**Curvature Sphere** will display the
curvature sphere.

The curvature sphere moves according to the movement of the *u*-indicator.
As mentioned earlier, the center of the curvature sphere is on the positive
direction of the normal vector, and is displayed as a square.
The left figure below shows a curvature sphere. But, when the tracing point
moves to the top of the curve which is flat, the curvature sphere becomes
bigger indicating the the curve is flat in the vicinity of the tracing
point.

If the tracing point moves further to an inflection point (*i.e.*, a
point whose second derivative is zero), the curvature sphere becomes
extremely large. Moving pass this inflection point and entering a sharp
turn, the curvature sphere is small, meaning that the curvature is large and
the curve has a sharp turn there.