Question

What is the name of the surface that is generated with linear interpolation of two curves?

• A. Bezier surface
• B. B-spline surface
• C. Planar surface
• D. Ruled surface

Hint:

A ruled surface has the property that through every point on the surface, there passes at least one straight line that lies entirely on the surface.
A classic physical example is a cylinder, cone, or hyperbolic paraboloid.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
This CAD/CAM and computer graphics question asks for the classification of a surface generated by linearly interpolating between two distinct boundary curves.

Step 2: Key Formula or Approach:
If we have two spatial curves, $C_1(u)$ and $C_2(u)$, parameterized by $u$, a surface $S(u, v)$ can be constructed by connecting corresponding points on the two curves with straight lines.
The mathematical equation for this linear interpolation is:
\[ S(u, v) = (1 - v) \cdot C_1(u) + v \cdot C_2(u) \]
where $v$ is the interpolation parameter ranging from $0$ to $1$.

Step 3: Detailed Explanation:

• A surface generated by sweeping a straight line through space along a prescribed path is called a ruled surface.

• In CAD modeling, when we connect two profile curves linearly, the straight lines connecting them are the "rulings" of the surface.

• Option A (Bezier surface) is a polynomial surface controlled by a grid of control points, not necessarily straight-line interpolation between two boundary curves.

• Option B (B-spline surface) offers local control using B-spline basis functions.

• Option C (Planar surface) is a flat surface, which is a very specific, limited sub-case of a ruled surface.

• Therefore, the general term for a surface generated by linear interpolation between two arbitrary space curves is a ruled surface.

Step 4: Final Answer:
The surface generated with linear interpolation of two curves is a ruled surface.