Determine the Characteristics of the Intersecting Projection Onto the Plane Containing Two Parallel Axes of Two Revolution Surfaces of Second-order

When representing objects, frequently encounter problems constructing the intersection of two revolution surfaces of second-order. Two revolution surfaces of second-order intersect in a quaternary rapids curve. If two revolution surfaces of second–order have parallel axes or one of them is a sphere, the projection of the intersection onto the plane containing their two axes is a parabolic curve. However, there has not been any research on determining the characteristics of the projection parabola of the intersection, but only use auxiliary surfaces to regulate the points on the intersection. The paper presents the method to accurately ascertain the characteristics of the projection parabola of the intersection including vertex, symmetric, focus and directrix. From this result, the parabola is completely defined, it can be drawn quickly without using auxiliary surfaces and even possible to determine imaginary points on the intersection parabola