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Homework 5 Problems

1) Given two 2D points (4,2) and (6,-2)

a) what parametric equation(s) represent
the line segment connecting these points?
x = at + b and y = ct + d
where t is the parameter
x = 2.0t + 4.0
y = -4.0t + 2.0
where 0.0 <= t <= 1.0

b) what is the slope/intercept equation for the line containing this line segment?
y = ax + b ; m=(y-y/x-x) =(2+2)/(4-6)= =-2 ;
y=-2x+10

c) what is the vector connecting the first point to the second point? [2,-4]

2) Is the polygon defined by the following points
(-1,0), (5,6), (9,1) and (4,-2) a concave or convex polygon?

Convex

 

3) Given the polygon from problem 2. If this polygon is concave, how would you change its 4th point to get a
convex polygon. If on the other hand the given polygon is convex, how would you change its 4th point to get a
concave polygon. You would want to move the fourth point up (beyond where a line between the 1st and 3rd points would be). (Example: (4,2))



4) Given a spline "span" specified by
x = f(t) = t2 +2.0 t + 2.0 and
y = g(t) = 2.0 t2 - 2.0t + 2.0 where 0.0 <= t <= 1.0

a) what are the points on the spline when t = 0.1, t = 0.3, t = 0.5, t= 0.7, and t = 0.9

t	.1	.3	.5	.7	.9
x	2.21	2.69	3.25	3.89	4.61
y	1.82	1.58	1.5	1.58	1.82

b) what is the order of this spline? quadratic
c) given another spline span specified by x = t + 5.0 and y = - 3.0 t + 2.0, what type of "continuity"
exists between these two spans?
When t=0, x=5, y=2
When t=1, x=6, y=-1
No continuity
d) what is the order of this second spline? linear

5) How many dimensions does a voxel have? How many unit voxels are needed to represent a 20 x 20 x 40 volume? Voxels have 3 dimensions. (20X20X40) units needed = 16000 units

6) Show a 5 x 3 pixel array (5 rows, 3 columns) containing a representation of the figure "3"

X	X	X
		X
	X	X
		X
X	X	X

7) Show an 8 segment polyline
approximation to a semi-circle
of radius 10 whose center is at
(5, -5). List all of the vertex
coordinates for this polyline.
A (–5,-5)
B (-4.2388, -1.1732)
C (-2.0711, 2.0711)
D (1.1732, 4.2388)
E (5, 5)
F (8.8268, 4.2388)
G (12.0711, 2.0711)
H (14.2388, -1.1732)
I (15,-5)

 

8) How would you define, in cartesian coordinates, a cylindrical surface with a radius of 2 and a length of 10, centered about the origin and the Z axis.

a) using a surface of revolution Create a line 10 units long [2,0,-5] to [2,0,5] . Rotate this 360 degrees around the z axis and it will create a cylindrical surface 10 units long, with a diameter of 4.
b) using an extruded surface Create a circle with radius 2 units, centered at [0,0,-5] on the z axis. Extrude this out 10 units in the positive direction to create a cylinder 10 units long, with a diameter of 4.
c) using a polygonal mesh Create a ‘circle’ as in ‘b)’, (determine how many sides you want for this, the greater number of sides, the smoother the surface), 10 units in the positive direction copy the circle, and connect the points to create the polygonal mesh.
d) using a bi-parametric surface This is determined by the order of the function, whether bi-cubic or bi-quadratic. Use directions from c) and define the span.