State whether the following statements are true or false. Justify your answer. (

Question

State whether the following statements are true or false. Justify your answer. (a) The refresh rate of a 512 times 512 frame buffer is approximately 21 frames/second, if the access time for each pixel is 200 nanoseconds. (b) 2-D rotations about the origin are non-commutative. (c) If a polygon lies on a plane Ax + By + Cz + D = 0, then N = (A, B, C) is normal to that plane. (d) The matrix [d 0 0 0 0 d 0 0 0 0 d 0 0 0 1 0] represents the perspective projection matrix on the plane z = d, where the centre of projection is (0, 0, 0). (e) A cubic Bezier curve cannot be drawn if the control points are located at the vertices of a rectangle.

Explanation / Answer

1.For individual access of pixel, i.e for one pixel its 200 * 10^-9 (nano) seconds, therefore for 512*512 pixels, 0.0524 seconds is required, which means refresh rate per second is =1/0.0524 which is ~ 19 frames/second.

so its false.

2. false

3. to check if a point is in plane or not we need to take dot product of the 2 line vectors. so (a,b,c) to origin(assuming since 2 start point is given) we have a.a+b.b+c.c which can never be zero, hence it is normal to plane and above it since dot product >0 TRUE

4.TRUE

A perspective transformation is determined by prescribing a  of projection and a view plane. The view plane is determined by its view reference point R0 and view plane normal N.

The plance z = d is parallel, to the xy plane. Thus the view plane normal vector N is the same as the normal vector K to the xy plance, that is N=K. Choosing the view reference point as R0(0,0,d), we can identify the parameters

N(n1,n2,n3) = (0,0,1)
R0(x0,y0,z0)=(0,0,d)

So,
d0 = n1x0 + n2y0 + n3z0 = d

and so the projection matrix is
PerK,R0 = d 0 0 0
0 d 0 0
0 0 d 0
0 0 1 0

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