# CS代写|计算机图形学作业代写computer graphics代考|CS4600

## CS代写|计算机图形学作业代写computer graphics代考|Odd and Even Functions

An odd function satisfies the condition:
$$f(-x)=-f(x)$$

where $x$ is located in a valid domain. Consequently, the graph of an odd function is symmetrical relative to the origin. For example, $\sin (\theta)$ is odd because
$$\sin (-\theta)=-\sin \theta$$
as illustrated in Fig. 3.6. Other odd functions include:
\begin{aligned} &f(x)=a x \ &f(x)=a x^3 \end{aligned}
An even function satisfies the condition:
$$f(-x)=f(x)$$
where $x$ is located in a valid domain. Consequently, the graph of an even function is symmetrical relative to the $f(x)$ axis. For example. $\cos \theta$ is even because
$$\cos (-\theta)=\cos \theta$$
as illustrated in Fig. 3.7. Other even functions include:
\begin{aligned} &f(x)=a x^2 \ &f(x)=a x^4 \end{aligned}

## CS代写|计算机图形学作业代写computer graphics代考|Units of Angular Measurement

The measurement of angles is at the heart of trigonometry, and today two units of angular measurement are part of modern mathematics: degrees and radians. The degree (or sexagesimal) unit of measure derives from defining one complete rotation as $360^{\circ}$. Each degree divides into $60 \mathrm{~min}$, and each minute divides into $60 \mathrm{~s}$. The number 60 has survived from Mesopotamian days and appears rather incongruous when used alongside today’s decimal system-nevertheless, it is still convenient to work with degrees even though the radian is a natural feature of mathematics.
The radian of angular measure does not depend upon any arbitrary constant, and is often defined as the angle created by a circular arc whose length is equal to the circle’s radius. And because the perimeter of a circle is $2 \pi r, 2 \pi$ rad correspond to one complete rotation. As $360^{\circ}$ corresponds to $2 \pi$ rad, 1 rad equals $180^{\circ} / \pi$, which is approximately $57.3^{\circ}$. The following relationships between radians and degrees are worth remembering:
\begin{aligned} \frac{\pi}{2}[\mathrm{rad}] & \equiv 90^{\circ}, & \pi[\mathrm{rad}] & \equiv 180^{\circ} \ \frac{3 \pi}{2}[\mathrm{rad}] & \equiv 270^{\circ}, & 2 \pi[\mathrm{rad}] \equiv 360^{\circ} . \end{aligned}
To convert $x^{\circ}$ to radians:
$$\frac{\pi x^{\circ}}{180}[\mathrm{rad}] .$$
To convert $x$ [rad] to degrees:
$$\frac{180 x}{\pi} \text { [degrees]. }$$
For those readers wishing to know the background to radians we need to use power series. We start with the power series for $\mathrm{e}^\theta, \sin \theta$ and $\cos \theta$ :
$$\begin{gathered} \mathrm{e}^\theta=1+\frac{\theta^1}{1 !}+\frac{\theta^2}{2 !}+\frac{\theta^3}{3 !}+\frac{\theta^4}{4 !}+\frac{\theta^5}{5 !}+\frac{\theta^6}{6 !}+\frac{\theta^7}{7 !}+\frac{\theta^8}{8 !}+\frac{\theta^9}{9 !}+\cdots \ \sin \theta=\theta-\frac{\theta^3}{3 !}+\frac{\theta^5}{5 !}-\frac{\theta^7}{7 !}+\frac{\theta^9}{9 !}+\cdots \ \cos \theta=1-\frac{\theta^2}{2 !}+\frac{\theta^4}{4 !}-\frac{\theta^6}{6 !}+\frac{\theta^8}{8 !}+\cdots . \end{gathered}$$
Euler proved that these three power series are related, and when $\theta=\pi, \sin \theta=0$, and $\cos \theta=-1$. Figure $4.1$ shows curves of the sine power series for $3,5,7$ and 9 terms, and when $\theta=2 \pi$, the graph reaches zero.

# 计算机图形学代考

## CS代写|计算机图形学作业代写computer graphics代考|Odd and Even Functions

$$f(-x)=-f(x)$$

$$\sin (-\theta)=-\sin \theta$$

$$f(x)=a x \quad f(x)=a x^3$$

$$f(-x)=f(x)$$

$$\cos (-\theta)=\cos \theta$$

$$f(x)=a x^2 \quad f(x)=a x^4$$

## CS代写|计算机图形学作业代写computer graphics代考|Units of Angular Measurement

$$\frac{\pi}{2}[\mathrm{rad}] \equiv 90^{\circ}, \quad \pi[\mathrm{rad}] \equiv 180^{\circ} \frac{3 \pi}{2}[\mathrm{rad}] \quad \equiv 270^{\circ}, 2 \pi[\mathrm{rad}] \equiv 360^{\circ} .$$

$$\frac{\pi x^{\circ}}{180}[\mathrm{rad}] .$$

$$\frac{180 x}{\pi} \text { [degrees]. }$$

$$\mathrm{e}^\theta=1+\frac{\theta^1}{1 !}+\frac{\theta^2}{2 !}+\frac{\theta^3}{3 !}+\frac{\theta^4}{4 !}+\frac{\theta^5}{5 !}+\frac{\theta^6}{6 !}+\frac{\theta^7}{7 !}+\frac{\theta^8}{8 !}+\frac{\theta^9}{9 !}+\cdots \sin \theta=\theta-\frac{\theta^3}{3 !}+\frac{\theta^5}{5 !}-\frac{\theta^7}{7 !}+\frac{\theta^9}{9 !}+$$

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