課程名稱︰計算機圖形
課程性質︰選修
課程教師：歐陽明
開課學院：電資學院
開課系所︰資工所、網媒所
考試日期（年月日）︰2014.11.24
考試時限（分鐘）：
試題 :
Interactive Computer Graphics Mid-term Exam, 24 Nov. ,2014
1. Visibility (15%)
Figure: http://i.imgur.com/jsUnuXU.png
(a) (5%) Painter's algorithm draws polygons from back to front. Give an
example where painter's algorithm fails.
(b) (5%) BSP tree is an algorithm to address the problem painter's
algorithm faces. Construct the BSP tree for the following figure. Use
face 1 as the root.
(c) (5%) What is the display sequence if the eye is placed in the position
before face 3 and 2, but at the back of face 5.
2. Transformation (20%)
Define R_x, R_y, R_z as the rotation about the origin along x, y, z axis,
respectively. Besides, let T(d) be a translation which moves a point with
displacement d. Please
(1) (5%) Determine a transformation to rotate an object about a fixed
point p along x axis (θ is the rotation angle).
(2) (5%) Determine a transformation to rotate an object along an arbitrary
axis (θ is the rotation angle, u is the axis direction, and p is its
center of mass).
(3) (5%) Consider a scene composed of many similar objects (may simply
with different size, location, orientation, etc.). If we want to
create the scene, we can define vertices for each object directly, or
define different transformation on objects of same types. We call such
transformations as instance transformation. What is the merit of
instance transformation?
(4) (5%) For an object with its center of mass on the origin, an instance
transformation can be defined as the product of a translation, rotate,
and scaling. That is, the object is scaled first, then rotated, then
translated into the desired location. Can we accomplish the same effect
by applying these three types of transformation in a different order?
3. Ray Tracing Techniques (20%)
(1) (3%) Why there will be shadows in a scene with Ray Tracing Algorithm?
(2) (4%) There are a large number of techniques for improving the realism
of ray traced images. Briefly describe how each of the techniques below
can modify the lighting calculation:
1. Texture Mapping
2. Environment Mapping
(3) (3%) Which of these techniques would be most appropriate for modeling
each of the following? And why?
1. A picture hung on a wall.
2. A marble statue (大理石雕像).
3. A new ball made of glass.
(4) (5%) Can you describe what's the drawbacks of Ray Tracing, as compared
to The Rendering Equation. Please give an example.
(5) (5%) So, in the previous case, how would you improve the Ray Tracing
algorithm?
4. Ray-Object Intersection in Ray Tracing (10%)
A ray starts from (2014, 11, 24) and passes through the point
(2015, 12, 25). Find the intersection between the ray and a sphere
centered at (2016, 11, 24) with a radius of 5. Also find the normalized
surface normal of the intersections.
5. 3D projective transformation into 2D (10%)
Consider the standard projective transformation:
P(x, y, z) = (-x/z, -y/z, -1),
which projects the points onto the projection plane z = -1. Consider a
circle in 3D space, defined by the following equation:
╭ x^2 + (z + 1/2)^2 = 1/4
C: <
╰ y = 1
(1) (5%) Draw a diagram to show this circle is projected to a parabola onto
the projection plane.
(2) (5%) What are the homogeneous coordinates of the points at infinity of
this parabola in 3D space?
(3) (5% Bonus, 加分題) Derive the equation of this parabola.
6. Bezier Curve (10%)
Figure: http://i.imgur.com/Yl9L5eE.png
Usually we will use Bezier curves of order 3. Suppose that we want to join
two Bezier curves of order 2 end-to-end, using the control points sequence
(P_0, P_1, P_2), and (P_2, P_3, P_4), respectively. Exactly what
conditions must be satisfied by these five points for the combined curve
to have C^1 parametric continuity at the point at which they are joined.
Prove your answer carefully by showing the continuity of the derivatives at
this point.
7. (5%) What is your term project for this semester? What are the
technical difficulties involved in the project? (You can refer to
the project listing).
8. (10%) In two dimensions, we can specify a line by the equation y = mx + b.
(for example, a line L passing through (0, 0) and (10, 10).
(a) (5%) Find an affine transformation to reflect two-dimensional
points about this line. For example, point (2, 3) and it's
"mirror image(鏡面反射)" about line L above is (3, 2), and it is
a reflection.
(b) (5%) Extend your results to reflection about a plane
(ax + by + cz + d = 0, where a, b, c are real numbers) in three
dimensions. For example, point (0, 0, 0) about the plane x + y + z = 2,
and it's reflection point is (4/3, 4/3, 4/3).