Question 1

Let H be the hyperplane through the points. Find a linear functional f and a real number d such that H = [f : d].
-$\left[ \begin{array} { l } 1 \ 1 \ 1 \end{array} \right] , \left[ \begin{array} { r } 3 \ - 1 \ 5 \end{array} \right] , \left[ \begin{array} { r } 4 \ 2 \ - 2 \end{array} \right]$

Accepted Answer

The answer of Let H be the hyperplane through the...

Question 2

Provide an appropriate response
-A five dimensional hypercube $C ^ { 5 }$ has how many 2 -faces ?

Accepted Answer

The answer of Provide an appropriate response
-A five dimensional hypercube...

Question 3

Provide an appropriate response
-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.

Accepted Answer

I is true because the sum of two convex sets is also convex, which follows from the definition of convexity. II is false because the number of vertices and edges in a polytope can vary depending on its shape and dimensionality; there's no rule that they must be equal in a four-dimensional polytope.

Question 4

Provide an appropriate response
-Which of the following statements are true?
I: If $S = \{ ( x , y ) : x - y = 0$ and $x \geq 0 \}$ and if $P$ is its profile, then conv $P = S$.
II: If $S = \{ ( x , y ) : x - y = 0$ and $0 \leq x \leq 5 \}$ and if $P$ is its profile, then conv $P = S$.

Accepted Answer

Statement II is true because the set $S$ is a line segment from (0,0) to (5,5), which is already convex, so its convex hull is itself. Statement I is false because $S$ is a ray starting from (0,0) extending to infinity, which is not bounded, and thus its convex hull would not be equal to $S$.

Question 5

Provide an appropriate response.
-If 2 Bézier curves are joined at the point $\mathbf { p } _ { 3 }$ what is necessary for $\mathrm { G } ^ { 1 }$ geometric continuity?

Accepted Answer

For $G^1$ geometric continuity, the tangent vectors at the joining point of two Bézier curves must point in the same direction, but they do not need to be of equal magnitude. This ensures a smooth transition between the curves without a visible corner.

Question 6

Provide an appropriate response.
-If a Bézier curve is translated, $\mathbf { x } ( \mathrm { t } ) + \mathbf { b }$, will the new curve always be a Bézier curve as well?

Accepted Answer

The answer of Provide an appropriate response.
-If a Bézier curve...

Question 7

Let H be the hyperplane through the points. Find a linear functional f and a real number d such that H = [f : d].
-$$\left[ \begin{array} { r } 
- 1 \
3
\end{array} \right] , \left[ \begin{array} { r } 
3 \
- 4
\end{array} \right]$$

Accepted Answer

The answer of Let H be the hyperplane through the...

Question 8

Provide an appropriate response.
-A quadratic Bézier curve is determined by 3 control points $\mathbf { p } _ { 0 } , \mathbf { p } _ { 1 }$, and $\mathbf { p } _ { 2 }$. The equation is $\mathbf { x } ( \mathrm { t } ) =$ $( 1 - t ) ^ { 2 } \mathbf { p } _ { 0 } + 2 t ( 1 - t ) \mathbf { p } _ { 1 } + \mathrm { t } ^ { 2 } \mathbf { p } _ { 2 }$. Construct the quadratic Bézier basis matrix $\mathrm { M } _ { \mathrm { B } }$ for $\mathbf { x } ( \mathrm { t } )$.

Accepted Answer

The answer of Provide an appropriate response.
-A quadratic Bézier curve...

Question 9

Provide an appropriate response
-Consider the set $S$ of points $\left[ \begin{array} { l } x \ y \end{array} \right]$ in $R ^ { 2 }$ such that $y = \frac { 1 } { x }$ and $x \geq \frac { 1 } { 2 }$. Are the sets $S$ and conv $S$ both closed?

Accepted Answer

The set $S$ is closed because it is the graph of a continuous function on a closed interval. The set conv $S$ is also closed because it is the convex hull of a closed set.

Question 10

Provide an appropriate response
-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.

Accepted Answer

Both statements I and II are true. For a convex set S, both the interior and the closure of S are also convex. The interior is convex because any line segment between two points in the interior remains entirely within the interior. The closure is convex because it includes all the limit points of the convex set, ensuring that any line segment between two points in the closure remains within the closure.

Question 11

Provide an appropriate response.
-Let $x ( t )$ be a Bézier curve and the tangent vector $x ^ { \prime } ( t )$ is computed. What does knowing that $\mathbf { x } ^ { \prime } ( 0 ) =$ $3 \left( \mathbf { p } _ { 1 } - \mathrm { p } _ { 0 } \right)$ tell you?

Accepted Answer

The answer of Provide an appropriate response.
-Let \(x ( t...

Question 12

Provide an appropriate response
-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.

Accepted Answer

I: A polytope is actually defined as the convex hull of a finite set of points, not the affine hull. II: An extreme point of a polytope P is a vertex of P, not any point in the convex hull of 2 vertices.

Question 13

Provide an appropriate response
-Which of the following statements are true?
I: The open ball $\mathrm { B } ( \mathbf { p } , \delta )$ is a convex set.
II: The convex hull of a compact set is compact.

Accepted Answer

The answer of Provide an appropriate response
-Which of the following...

Question 14

Provide an appropriate response.
-How many control points are needed for a cubic Bézier curve?

Accepted Answer

The answer of Provide an appropriate response.
-How many control points...

Question 15

Provide an appropriate response
-A four dimensional simplex $S ^ { 4 }$ has how many 2 -faces?

Accepted Answer

A four-dimensional simplex, also known as a 5-cell or pentachoron, has 10 2-faces. The formula for calculating the number of k-faces in an n-simplex is given by the binomial coefficient $C(n+1, k+1)$, so for 2-faces in a 4-simplex, it's $C(5, 3) = 10$.

Question 16

Let H be the hyperplane through the points. Find a linear functional f and a real number d such that H = [f : d].
-$ \left[\begin{array}{l}1 \ 0 \ 0 \ 1\end{array}\right],\left[\begin{array}{l}2 \ 3 \ 1 \ 2\end{array}\right],\left[\begin{array}{r}-1 \ 1 \ 2 \ 1\end{array}\right],\left[\begin{array}{r}3 \ 2 \ -1 \ 1\end{array}\right] $

Accepted Answer

The answer of Let H be the hyperplane through the...

Question 17

Provide an appropriate response.
-If 2 Bézier curves are joined at the point $\mathbf { p } _ { 3 }$ what is necessary for $\mathrm { C } ^ { 1 }$ parametric continuity?

Accepted Answer

The answer of Provide an appropriate response.
-If 2 Bézier curves...

Quiz 8: The Geometry of Vector Spaces

Provide an appropriate response -A five dimensional hypercube $C ^ { 5 }$ has how many 2 -faces ?

Provide an appropriate response -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.

Provide an appropriate response -Which of the following statements are true? I: If $S = \{ ( x , y ) : x - y = 0$ and $x \geq 0 \}$ and if $P$ is its profile, then conv $P = S$ . II: If $S = \{ ( x , y ) : x - y = 0$ and $0 \leq x \leq 5 \}$ and if $P$ is its profile, then conv $P = S$ .

Provide an appropriate response. -If 2 Bézier curves are joined at the point $\mathbf { p } _ { 3 }$ what is necessary for $\mathrm { G } ^ { 1 }$ geometric continuity?

Provide an appropriate response. -If a Bézier curve is translated, $\mathbf { x } ( \mathrm { t } ) + \mathbf { b }$ , will the new curve always be a Bézier curve as well?

Let H be the hyperplane through the points. Find a linear functional f and a real number d such that H = [f : d]. - $\left[ \begin{array} { r } - 1 \\3\end{array} \right] , \left[ \begin{array} { r } 3 \\- 4\end{array} \right]$

Provide an appropriate response -Consider the set $S$ of points $\left[ \begin{array} { l } x \\ y \end{array} \right]$ in $R ^ { 2 }$ such that $y = \frac { 1 } { x }$ and $x \geq \frac { 1 } { 2 }$ . Are the sets $S$ and conv $S$ both closed?

Provide an appropriate response -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.

Provide an appropriate response. -Let $x ( t )$ be a Bézier curve and the tangent vector $x ^ { \prime } ( t )$ is computed. What does knowing that $\mathbf { x } ^ { \prime } ( 0 ) =$ $3 \left( \mathbf { p } _ { 1 } - \mathrm { p } _ { 0 } \right)$ tell you?

Provide an appropriate response -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.

Provide an appropriate response -Which of the following statements are true? I: The open ball $\mathrm { B } ( \mathbf { p } , \delta )$ is a convex set. II: The convex hull of a compact set is compact.

Provide an appropriate response. -How many control points are needed for a cubic Bézier curve?

Provide an appropriate response -A four dimensional simplex $S ^ { 4 }$ has how many 2 -faces?

Provide an appropriate response. -If 2 Bézier curves are joined at the point $\mathbf { p } _ { 3 }$ what is necessary for $\mathrm { C } ^ { 1 }$ parametric continuity?

Quiz 8: The Geometry of Vector Spaces

Provide an appropriate response -A five dimensional hypercube C5C ^ { 5 }C5 has how many 2 -faces ?

Provide an appropriate response -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.

Provide an appropriate response. -If 2 Bézier curves are joined at the point p3\mathbf { p } _ { 3 }p3​ what is necessary for G1\mathrm { G } ^ { 1 }G1 geometric continuity?

Provide an appropriate response. -If a Bézier curve is translated, x(t) +b\mathbf { x } ( \mathrm { t } ) + \mathbf { b }x(t) +b , will the new curve always be a Bézier curve as well?

Let H be the hyperplane through the points. Find a linear functional f and a real number d such that H = [f : d]. - [−13],[3−4]\left[ \begin{array} { r } - 1 \\3\end{array} \right] , \left[ \begin{array} { r } 3 \\- 4\end{array} \right][−13​],[3−4​]

Provide an appropriate response -Consider the set SSS of points [xy]\left[ \begin{array} { l } x \\ y \end{array} \right][xy​] in R2R ^ { 2 }R2 such that y=1xy = \frac { 1 } { x }y=x1​ and x≥12x \geq \frac { 1 } { 2 }x≥21​ . Are the sets SSS and conv SSS both closed?

Provide an appropriate response -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.

Provide an appropriate response -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.

Provide an appropriate response -Which of the following statements are true? I: The open ball B(p,δ) \mathrm { B } ( \mathbf { p } , \delta ) B(p,δ) is a convex set. II: The convex hull of a compact set is compact.

Provide an appropriate response. -How many control points are needed for a cubic Bézier curve?

Provide an appropriate response -A four dimensional simplex S4S ^ { 4 }S4 has how many 2 -faces?

Provide an appropriate response. -If 2 Bézier curves are joined at the point p3\mathbf { p } _ { 3 }p3​ what is necessary for C1\mathrm { C } ^ { 1 }C1 parametric continuity?

Provide an appropriate response -A five dimensional hypercube $C ^ { 5 }$ has how many 2 -faces ?

Provide an appropriate response. -If 2 Bézier curves are joined at the point $\mathbf { p } _ { 3 }$ what is necessary for $\mathrm { G } ^ { 1 }$ geometric continuity?

Provide an appropriate response. -If a Bézier curve is translated, $\mathbf { x } ( \mathrm { t } ) + \mathbf { b }$ , will the new curve always be a Bézier curve as well?

Let H be the hyperplane through the points. Find a linear functional f and a real number d such that H = [f : d]. - $\left[ \begin{array} { r } - 1 \\3\end{array} \right] , \left[ \begin{array} { r } 3 \\- 4\end{array} \right]$

Provide an appropriate response -Consider the set $S$ of points $\left[ \begin{array} { l } x \\ y \end{array} \right]$ in $R ^ { 2 }$ such that $y = \frac { 1 } { x }$ and $x \geq \frac { 1 } { 2 }$ . Are the sets $S$ and conv $S$ both closed?

Provide an appropriate response -Which of the following statements are true? I: The open ball $\mathrm { B } ( \mathbf { p } , \delta )$ is a convex set. II: The convex hull of a compact set is compact.

Provide an appropriate response -A four dimensional simplex $S ^ { 4 }$ has how many 2 -faces?

Provide an appropriate response. -If 2 Bézier curves are joined at the point $\mathbf { p } _ { 3 }$ what is necessary for $\mathrm { C } ^ { 1 }$ parametric continuity?