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Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the barycentric coordinates of p with respect to the affinely independent set of points that precedes it.
1)
1
1, 3
5, 0
6, p=
11
2
-1
2
1)
A)
2, 5
2, -7
2
B)
1, 3
2, -3
2
C)
(-9, 3, 7)
D)
3
2, -3
2, 1
Provide an appropriate response.
2)
Suppose {v1, v2, v3} is a basis for
3. Which of the following statements are true?
I: Span {v2-v1, v3-v1} is a plane in
3.
II: Aff {v1, v2, v3} is the plane through v1, v2, and v3.
2)
A)
Both I and II
B)
I
C)
Neither I or II
D)
II
3)
´
Let x(t) be a Bezier curve and the tangent vector x(t) is computed. What does knowing that x(0) =
3(p1-p0) tell you?
3)
A)
The tangent vector points in the direction from p0 to -p1 and its length is 3 times the length of
p1.
B)
The tangent vector points in the direction from p0 to p1 and its length is 3 times the length of
p1-p0.
C)
The tangent vector points in the direction from p0 to p1 and its length is 3.
D)
The tangent vector points in the direction from p1 to p0 and its length is 3 times the length of
p1-p0.
4)
´ ´
If a Bezier curve is translated, x(t) +b, will the new curve always be a Bezier curve as well?
4)
A)
´
No, it will almost never be a Bezier curve. There are only a few choices that would work.
B)
´
No, it will never be a Bezier curve.
C)
´
Yes, it will always be a Bezier curve.
D)
´
No, but it will almost always be a Bezier curve with only a few exceptions.
Answer the question.
5)
Which of the following statements are true?
I: If A and B are convex sets then A + B is convex.
II: A four dimensional polytope always has the same number of vertices and edges.
5)
A)
I
B)
II
C)
Neither I or II
D)
Both I and II
Provide an appropriate response.
6)
p =1
2v1+1
8v2+1
8v3+1
4v4 and 2v1+v2- 2v3-v4=0
Use the procedure in the proof of Caratheodory's Theorem to express p as a convex combination of
three of the vi's.
6)
A)
p =1
4v1+3
8v2+3
8v4
B)
p =1
8v2+3
8v3+1
2v4
C)
p =5
8v1+3
16v2+3
16v4
D)
It cannot be done using only 3 vi's.
Let H be the hyperplane through the points. Find a linear functional f and a real number d such that H = [f : d].
7)
1
0
0
1
,
2
3
1
2
,
-1
1
2
1
,
3
2
-1
1
7)
A)
f(x1, x2, x3, x4) = -5x1+ 2x2- 6x3+ 5x4, d = 0
B)
f(x1, x2, x3, x4) = 2x2-x3+ 5x4, d = 5
C)
f(x1, x2, x3, x4) = 5x1+ 6x2+ 2x3+ 5x4, d = 10
D)
f(x1, x2, x3, x4) =x1- 2x2- 6x3+ 13x4, d = 12
Provide an appropriate response.
8)
Pick a set S of four distinct points in
3 such that aff S is the plane 3x1+x2-3x3=12.
8)
A)
S=2
12
1, -1
6
-1, 3
6
0 , 0
0
0
B)
S=3
1
-3, 6
2
-6, 0
12
0 , 1
12
1
C)
S=0
9
1, 1
6
1, 2
18
0 , 0
12
0
D)
S=0
15
1, 1
6
-1, -2
18
0 , 0
12
0
9)
Which of the following statements are true?
I: If S is a nonempty set, then conv S
S.
II: If S and T are convex sets, then S
T is also convex.
9)
A)
I
B)
Neither I or II
C)
II
D)
Both I and II
Determine if the set of points is affinely dependent. If so, construct an affine dependence relation for the points.
10)
4
4, -12
-12 , 0
0
10)
A)
The set is affinely dependent. 3 4
4+-12
-12 - 4 0
0=0
0
B)
The set is affinely dependent. 3 1
1+-3
-3-0
0=0
0
C)
The set is affinely independent.
D)
The set is affinely dependent. 3 4
4--12
-12 + 4 0
0=0
0
Provide an appropriate response.
11)
Which of the following statements are true?
I: 2v1+ 2v2- 3v3 is an affine combination of the 3 vectors.
II: The affine hull of two distinct points is a plane.
III: If S = {x}, then aff S = {x}.
IV: If a set of vectors in
n is linearly independent, then every vector in
n can be written as an
affine combination of these vectors.
11)
A)
I , III, and IV
B)
I and II
C)
I and III
D)
II and IV
Find the barycentric coordinates of p with respect to the affinely independent set of points that precedes it.
12)
1
-1
2
1
,
2
1
0
1
,
1
2
-2
0
, p=
-1
-8
10
2
12)
A)
(4, 0, -3)
B)
(2, -2, 1)
C)
(3, -1, -1)
D)
(4, -2, -1)
Determine if the vector p is in Span S or aff S.
13)
Let v1=
2
1
3
-2
, v2=
3
-1
0
4
, v3=
5
4
-1
-2
, p=
2
-3
7
2
, and S = {v1, v2, v3}. It can be shown that S is linearly
independent.
13)
A)
p
span S and p
aff S
B)
p
span S and p
aff S
C)
p
span S and p
aff S
D)
p
span S and p
aff S
Use the barycentric coordinates with respect to S to determine if the point p is inside, outside, on a face, or on the edge of
conv S which is a tetrahedron.
14)
S = {v1, v2, v3, v4,}
Barycentric coordinates: 1
4, 3
8, 1
4, 1
8
14)
A)
Face
B)
Edge
C)
Outside
D)
Inside
Answer the question.
15)
A four dimensional simplex S4 has how many 2-faces?
15)
A)
5
B)
10
C)
4
D)
15
16)
A five dimensional hypercube C5 has how many 2-faces ?
16)
A)
24
B)
40
C)
80
D)
32
Sketch the graph of the convex hull of S.
5
17)
In
2, let S =0
5x
0: 0 x 3
17)
A)
B)
C)
D)
Provide an appropriate response.
18)
[(aff A) (aff B)]
(A B)
What property must the set (A B) have if the above statement is true?
18)
A)
There is nothing special about (A B). The statement is always true.
B)
(A B) has to be affine.
C)
A
B so that (A B) = A.
D)
(A B) must contain collinear points.
Answer the question.
19)
Consider the affinely independent set S = {v1, v2, v3} in
2. These points form a triangular region.
If the barycentric coordinates of a point p are all positive (+, +, +), then where is p with respect to
the triangle?
19)
A)
Outside
B)
Inside
C)
Edge
D)
Vertex
20)
Which of the following statements are true?
I: If S = {(x, y): x - y = 0 and x 0} and if P is its profile, then conv P = S.
II: If S = {(x, y): x - y = 0 and 0 x 5} and if P is its profile, then conv P = S.
20)
A)
I
B)
II
C)
Both I and II
D)
Neither I or II
Provide an appropriate response.
21)
´
How many control points are needed for a cubic Bezier curve?
21)
A)
5
B)
4
C)
3
D)
2
Answer the question.
22)
Which of the following statements are true?
I: A linear transformation from
to
n is called a linear functional.
II: The convex hull of a closed set is closed.
22)
A)
I
B)
Both I and II
C)
II
D)
Neither I or II
23)
Let int S be the set of all interior points of S, and let cl S be the closure of S (S the set of all
boundary points of S). Which of the following statements are true?
I: If S is convex, then int S is convex.
II: If S is convex, then cl S is convex.
23)
A)
Neither I or II
B)
Both I and II
C)
II
D)
I
Write y as an affine combination of the other points listed.
24)
v1=5
4, v2=3
5, v3=11
5, y=9
10
24)
A)
y=v1- 2v2+ 2v3
B)
y= 4v1- 2v2-v3
C)
y= -5v1+ 4v2+ 2v3
D)
y=5v1+ 3v2- 7v3
25)
v1=1
1
2, v2= 0
4
-2, v3= 1
-5
1 , y=6
-38
18
25)
A)
y=7v1+2v2- 5v3
B)
y=2v1- 5v2+ 4v3
C)
y=4v1+2v2- 5v3
D)
y= -2v1- 2v2+ 5v3
Sketch the graph of the convex hull of S.
8
26)
In
2, S is the set of points x
y where y =x2 and x 0
26)
A)
B)
C)
D)
Provide an appropriate response.
27)
Let (pos S) be the set of all positive combinations of the points of S. Which of the following
statements are true?
I: (pos S)
(aff S) = conv S
II: If a unique linear combination of points is both positive and affine, then it must be convex.
27)
A)
I
B)
Neither I or II
C)
Both I and II
D)
II
Use the barycentric coordinates with respect to S to determine if the point p is inside, outside, on a face, or on the edge of
conv S which is a tetrahedron.
28)
S = {v1, v2, v3, v4,}
Barycentric coordinates: (4, 1, 0, -4)
28)
A)
Inside
B)
Edge
C)
Outside
D)
Face
Provide an appropriate response.
29)
´
If 2 Bezier curves are joined at the point p3 what is necessary for G1 geometric continuity?
29)
A)
Both tangent vectors at p3 need to be equal.
B)
Both tangent vectors at p3 need to point in the same direction.
C)
The second derivative at p3 needs to equal 0.
D)
p3 just needs to be a control point for both curves.
30)
Let p0, p1, and p2 be points in
n and define f0(t) = (1 - t)p0+ tp1,f1(t) = (1 - t)p1+ tp2, and
g(t) = (1 - t)f0(t) + tf1(t) for 0 t 1.
Find g1
2 in terms of the three points.
30)
A)
g1
2=1
4p0+1
2p1+1
4p2
B)
g1
2=1
2p0+1
2p1
C)
g1
2=3
2p0+1
2p1-1
2p2
D)
g1
2=9
16p0+3
8p1+1
16p2
Determine if the vector p is in Span S or aff S.
31)
Let v1=
2
1
3
-2
, v2=
3
-1
0
4
, v3=
5
4
-1
-2
, p=
5
-3
5
3
, and S = {v1, v2, v3}. It can be shown that S is linearly
independent.
31)
A)
p
span S and p
aff S
B)
p
span S and p
aff S
C)
p
span S and p
aff S
D)
p
span S and p
aff S
Provide an appropriate response.
32)
´
If 2 Bezier curves are joined at the point p3 what is necessary for C1 parametric continuity?
32)
A)
p3 just needs to be a control point for both curves.
B)
Both tangent vectors at p3 need to have the same magnitude and direction.
C)
The second derivative at p3 needs to equal 0.
D)
Both tangent vectors at p3 need to point in the same direction.
33)
Which of the following statements are true?
I: Suppose f:
n
m is a linear transformation and S is a convex subset of
n. It follows that the
set of images f(S) is a convex subset of
m.
II: If A
B, then conv A
B.
33)
A)
I
B)
Neither I or II
C)
Both I and II
D)
II
Use the barycentric coordinates with respect to S to determine if the point p is inside, outside, on a face, or on the edge of
conv S which is a tetrahedron.
34)
S = {v1, v2, v3, v4,}
Barycentric coordinates: 0, 0, 1
2, 1
2
34)
A)
Face
B)
Inside
C)
Outside
D)
Edge
Answer the question.
35)
Consider the affinely independent set S = {v1, v2, v3} in
2. These points form a triangular region.
If the barycentric coordinates of a point p are (+, 0, +), then
where is p with respect to the triangle?
35)
A)
Inside
B)
Vertex
C)
Edge
D)
Outside
36)
If {v1, v2} in
n is affinely dependent, what is known about the 2 points?
36)
A)
v1=v2
B)
v1= cv2 where c
C)
{v1, v2} is linearly independent.
D)
{v2-v1} is linearly dependent.
37)
Let p be a point in the interior of
abc with barycentric coordinates 1
2, 3
8, 1
8. What is the area of
pbc with respect to the area of
abc?
37)
A)
3
8· (area of
abc)
B)
1
2· (area of
abc)
C)
1
8· (area of
abc)
D)
Not enough information to determine.
38)
Which of the following statements are true?
I: If F1 and F2 are 5 dimensional flats in
7, then the dimension of F1F2 could have a dimension
of 6.
II: If F1 and F2 are strictly separated, then F1F2=.
38)
A)
Both I and II
B)
Neither I or II
C)
I
D)
II
39)
Let H be the hyperplane through the three points 1
1
1, 3
3
2, 2
5
-2. Is the point v=8
1
2 on the same
side of H as the origin? Justify your answer.
39)
A)
Yes, because n · v < 3.
B)
No, because n · v > 3.
C)
No, because n · v > 0.
D)
Yes, because n · v <11.
40)
Which of the following statements are true?
I: The open ball B(p, ) is a convex set.
II: The convex hull of a compact set is compact.
40)
A)
I
B)
Neither I or II
C)
II
D)
Both I and II
Provide an appropriate response.
41)
´
´
A quadratic Bezier curve is determined by 3 control points p0, p1, and p2. The equation is x(t) =
(1 - t)2p0+ 2t(1 - t)p1+t2p2. Construct the quadratic Bezier basis matrix MB for x(t).
41)
A)
1-2 1
0 2 -2
0 0 1
B)
1 1
2-2
0 1
C)
12 2
02-2
00 1
D)
10 1
01-1
00 1
Use the barycentric coordinates with respect to S to determine if the point p is inside, outside, on a face, or on the edge of
conv S which is a tetrahedron.
42)
S = {v1, v2, v3, v4,}
Barycentric coordinates: 3
8, 0, 1
8, 1
2
42)
A)
Edge
B)
Outside
C)
Inside
D)
Face
Determine whether the point p is in the convex hull of S.
43)
S = {v1, v2, v3, v4}
v1=2
2
1, v2=0
3
-1, v3=1
8
-6, v4=-1
2
4, p =1
3
0
43)
A)
p
conv S. p =1
2v1+1
4v2+1
4v3
B)
p
conv S. p =1
2v1+1
4v2+1
8v3+1
8v4
C)
p
conv S.
D)
p
conv S. p = 2v1+ 3v2- 2v3- 2v4
Answer the question.
44)
Which of the following statements are true for the set S={v1, . . . , vk} in
n?
I: If S is affinely independent, then a point p in
n cannot have any barycentric coordinates
determined by S that are equal to 0.
II: If S is affinely independent, then a point p in aff S has a unique representation as an affine
combination of v1, ... , vk.
44)
A)
Neither I or II
B)
Both I and II
C)
I
D)
II
Let H be the hyperplane through the points. Find a linear functional f and a real number d such that H = [f : d].
45)
1
1
1, 3
-1
5 , 4
2
-2
45)
A)
f(x1, x2, x3) = -2x1+ 9x2+x3, d = 8
B)
f(x1, x2, x3) = 8x1+ 2x2+ 4x3, d = 14
C)
f(x1, x2, x3) =x1+ 9x2+ 4x3, d = 14
D)
f(x1, x2, x3) =x1+x2+5
2x3, d =9
2
Provide an appropriate response.
46)
Which of the following statements are true?
I: Suppose f :
n
m is a linear transformation and S is an affine subset of
n. It follows that the
set of images f(S) is an affine subset of
m.
II: If A
B, then aff A
B.
46)
A)
II
B)
Both I and II
C)
Neither I or II
D)
I
Let H be the hyperplane through the points. Find a linear functional f and a real number d such that H = [f : d].
47)
-1
3 , 2
-3
47)
A)
f(x1, x2) = -6x1+3x2, d =15
B)
f(x1, x2) = -x1+ 3x2, d = 10
C)
f(x1, x2) =6x1+3x2, d =3
D)
f(x1, x2) =2x1- 3x2, d = -11
Provide an appropriate response.
48)
v1=6
2
5, v2=6
3
-1, v3=6
4
3 and S = {v1, v2, v3}. Aff S is a plane in
3. Give its equation.
48)
A)
13x +14y =17
B)
x =6
C)
x + y + z = 0
D)
18x + 9y + 7z = 0
Determine if the set of points is affinely dependent. If so, construct an affine dependence relation for the points.
49)
2
4
-1, -4
-8
3, 0
1
10 , -8
-15
17
49)
A)
The set is affinely dependent. 2 2
4
-1+ 3 -4
-8
3+0
1
10 --8
-15
17 =0
0
0
B)
The set is affinely dependent. 2 2
4
-1- 3 -4
-8
3-0
1
10 +-8
-15
17 =0
0
0
C)
The set is affinely dependent. 5 2
4
-1- 3 -4
-8
3-0
1
10 --8
-15
17 =0
0
0
D)
The set is affinely independent.
Determine whether the point p is in the convex hull of S.
50)
S = {v1, v2, v3, v4}
v1=
2
0
5
1
, v2=
1
1
-1
-3
, v3=
3
2
0
-4
, p =
0
-1
4
2
,
50)
A)
p
conv S. p =v1+v2-v3
B)
p
conv S. p =3
8v1+1
4v2+3
8v3
C)
p
conv S.
D)
p
conv S. p = 2v1+ 3v2- 4v3
Answer the question.
51)
Which of the following statements are true?
I: A polytope is the affine hull of a finite set of points.
II: An extreme point of a polytope P is any point in the convex hull of 2 vertices.
51)
A)
II
B)
Both I and II
C)
Neither I or II
D)
I
52)
How many points need to be in a set in
6 to guarantee that the set is affinely dependent?
52)
A)
8
B)
6
C)
9
D)
7
Determine if the vector p is in Span S or aff S.
53)
Let v1=
2
1
3
-2
, v2=
3
-1
0
4
, v3=
5
4
-1
-2
, p=
10
12
-6
-8
, and S = {v1, v2, v3}. It can be shown that S is linearly
independent.
53)
A)
p
span S and p
aff S
B)
p
span S and p
aff S
C)
p
span S and p
aff S
D)
p
span S and p
aff S
Provide an appropriate response.
54)
Which of the following statements are true?
I: If A
B, then A
conv B.
II: If A
B, then conv A
conv B.
54)
A)
Neither I or II
B)
I
C)
Both I and II
D)
II
Answer the question.
55)
~ ~ ~ ~~~
Which of the following statements are true if p is a point on the line through a and b?
I: p is an affine combination a and b.
II: a, b, and p are linearly independent.
III: The determinant of [a b p] is 0.
55)
A)
II
B)
I and II
C)
I and III
D)
None are true.
56)
Consider the set S of points x
y in
2 such that y =1
x and x 1
2. Are the sets S and conv S both
closed?
56)
A)
No. Conv S is closed but S is not.
B)
Yes. They are both closed sets.
C)
No. Neither set is closed.
D)
No. S is closed but conv S is not.
57)
Which of the following statements are true for the set S={v1, . . . , vk} in
n?
I: {v2-v1, . . . , vk-v1} is linearly dependent if and only if S is affinely dependent.
II: If S is affinely independent and a point p in
n has all positive barycentric coordinates
determined by S, then p is not in aff S.
57)
A)
Both I and II
B)
I
C)
II
D)
Neither I or II
Answer Key
Testname: C8
Answer Key
Testname: C8
