• :00Days
• :00Hours
• :00Mins
• 00Seconds
A new era for learning is coming soon

Suggested languages for you:

Americas

Europe

Q11E

Expert-verified
Found in: Page 437

### Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

# Question: In Exercises 11 and 12, mark each statement True or False. Justify each answer.11.a. The cubic Bezier curve is based on four control points.b. Given a quadratic Bezier curve $${\mathop{\rm x}\nolimits} \left( t \right)$$ with control points $${{\mathop{\rm p}\nolimits} _0},{{\mathop{\rm p}\nolimits} _1},$$ and $${{\mathop{\rm p}\nolimits} _2}$$, the directed line segment $${{\mathop{\rm p}\nolimits} _1} - {{\mathop{\rm p}\nolimits} _0}$$ (from $${{\mathop{\rm p}\nolimits} _0}$$ to $${{\mathop{\rm p}\nolimits} _1}$$) is the tangent vector to the curve at $${{\mathop{\rm p}\nolimits} _0}$$.c. When two quadratic Bezier curves with control points $$\left\{ {{{\mathop{\rm p}\nolimits} _0},{{\mathop{\rm p}\nolimits} _1},{{\mathop{\rm p}\nolimits} _2}} \right\}$$ and $$\left\{ {{{\mathop{\rm p}\nolimits} _2},{{\mathop{\rm p}\nolimits} _3},{{\mathop{\rm p}\nolimits} _4}} \right\}$$ are joined at $${{\mathop{\rm p}\nolimits} _2}$$, the combined Bezier curve will have $${C^1}$$ continuity at $${{\mathop{\rm p}\nolimits} _2}$$if$${{\mathop{\rm p}\nolimits} _2}$$ is the midpoint of the line segment between $${{\mathop{\rm p}\nolimits} _1}$$ and $${{\mathop{\rm p}\nolimits} _3}$$.

1. The statement is True.
2. The statement is False.
3. The statement is True.
See the step by step solution

## Step 1: Determine whether the given statement is True or False

a)

The cubic Bezier curve is determined by four control points. The standard form is shown below:

$${\bf{x}}\left( t \right) = {\left( {1 - t} \right)^3}{{\bf{p}}_0} + 3t{\left( {1 - t} \right)^2}{{\bf{p}}_1} + 3{t^2}\left( {1 - t} \right){{\bf{p}}_2} + {t^3}{{\bf{p}}_3}$$

Thus, the given statement (a) is True.

## Step 2: Determine whether the given statement is True or False

b)

Recall that the tangent vector at $${{\mathop{\rm p}\nolimits} _0}$$, for instance, from $${{\mathop{\rm p}\nolimits} _0}$$ to $${{\mathop{\rm p}\nolimits} _1}$$, and it is twice as long as the segment from $${{\mathop{\rm p}\nolimits} _0}$$ to $${{\mathop{\rm p}\nolimits} _1}$$.

Thus, the given statement (b) is False.

## Step 3: Determine whether the given statement is True or False

c)

It is true that when two quadratic Bezier curve with control points $$\left\{ {{{\mathop{\rm p}\nolimits} _0},{{\mathop{\rm p}\nolimits} _1},{{\mathop{\rm p}\nolimits} _2}} \right\}$$and $$\left\{ {{{\mathop{\rm p}\nolimits} _2},{{\mathop{\rm p}\nolimits} _3},{{\mathop{\rm p}\nolimits} _4}} \right\}$$ then $${{\mathop{\rm p}\nolimits} _2}$$ is the midpoint of the line segment from $${{\mathop{\rm p}\nolimits} _1}$$ to $${{\mathop{\rm p}\nolimits} _3}$$.

The Bezier curve has $${C^1}$$ continuity at $${{\mathop{\rm p}\nolimits} _2}$$ when joined at $${{\mathop{\rm p}\nolimits} _2}$$, then $${{\mathop{\rm p}\nolimits} _2}$$ is the midpoint of the line segment from $${{\mathop{\rm p}\nolimits} _1}$$ to $${{\mathop{\rm p}\nolimits} _3}$$.

Thus, the given statement (c) is True.