Q1: Why aren’t polygonal models always sufficient for 3D graphics?
A1: Because polygons are flat; smooth curved shapes require many small polygons, which are inefficient and hard to manage.
Q2: What is the goal of implicit and parametric representations?
A2: To mathematically represent curves and surfaces smoothly using equations or parameterized functions instead of flat polygons.
Q3: What is an implicit representation?
A3: A curve or surface defined by an equation f(x,y,z)=0, describing all points that satisfy it.
Q4: Give examples of implicit equations.
Line: ax+by+c=0
Circle: x2+y2-r2=0
Sphere x2+y2+z2=0
Q5: What is the advantage of implicit forms?
A5: Compact mathematical definition, easy for analytical geometry and collision detection.
Q6: What is a major disadvantage of implicit forms in rendering?
A6: They’re difficult to display directly because they don’t provide explicit vertex coordinates.
Q7: How is the surface normal found for an implicit surface?
The gradient of f(x,y,z)
Q8: What does the gradient represent geometrically?
A8: The direction perpendicular to the surface — used for lighting and shading calculations.
Q9: What is a parametric representation?
A9: A function that generates points using one or more parameters: x=x(t), y=y(t), z=z(t)
Q10: Give the parametric equations of a circle.
x(t)=rcos(t), y(t)=rsin(t)
Q11: Give the parametric equations of a sphere.
x(u,v)=rcos(u)sin(v),y(u,v)=rsin(u)sin(v),z(u,v)=rcos(v)
Q12: Why are parametric forms better for rendering?
A12: They’re easy to sample at discrete parameter intervals and convert into vertices for drawing.
Q13: What is unit parameterization?
A13: Parameter t ranges from 0 to 1, representing the start and end of the curve.
Q14: What is arc-length parameterization?
A14: The parameter equals the physical distance along the curve — equal parameter steps correspond to equal distances.
Q15: Which parameterization is typically used in graphics?
A15: Unit parameterization, because it’s simpler to compute and implement.
Q16: What are piecewise curves?
A16: Curves made by connecting multiple simple curve segments smoothly.
Q17: Why use piecewise curves?
A17:
Each segment is simple to compute.
Together, they can form complex shapes (like car outlines or font curves).
Q18: What curve families use piecewise representations?
A18: Bézier curves, B-splines, and NURBS.
Q19: What does “continuity” mean for curves?
A19: How smoothly curve segments connect to each other.
Q20: What is C⁰ (parametric) continuity?
A20: The curve segments meet (no gaps).f1(1)=f2(0)
Q21: What is C¹ continuity?
A21: Tangents match at the junction (same direction and speed). f1′(1)=f2′(0)
Q22: What is C² continuity?
A22: The curvature is continuous (no sudden changes in bending).
Q23: What is geometric (Gⁿ) continuity?
A23: Only the direction of tangents and curvature match — not necessarily their magnitude.
Q24: How does geometric continuity differ from parametric continuity?
A24:
Cⁿ requires equal derivative magnitudes.
Gⁿ only requires proportional directions (visual smoothness).
Every C¹ curve is G¹, but not every G¹ curve is C¹.
