Course: 12.13 项目:极地坐标和参数等量

Section outline

Select section General

Collapse Expand
General

Collapse all Expand all
- Select activity 新闻通告
  
  新闻通告 Forum
Select section 项目:极地坐标和参数等量

Collapse Expand
项目:极地坐标和参数等量
A cycloid is the locus of a point on a circle as the circle "rolls" along a flat surface.
::环状物是圆形圆形上一个点的极点,即平坦表面的圆形“滚动”。

A hypocycloid is the locus of a point on the circumference of a circle that rolls without slipping inside the circumference of another circle.
::低环体是圆圈环绕点的中心点,环绕的圆圈不滑入另一圆圈的环圈。

For this project, we will investigate the parametric equations of the hypocycloid and the impact of changing the ratio of the radii of the two circles. We'll define our variables as follows:
::对于这个项目,我们将调查低环体的参数方程以及改变两个圆圈的射线比的影响。

Our hypocycloid is a curve traced by a fixed point $\begin{aligned} P \end{aligned}$ on a circle $\begin{aligned} C \end{aligned}$ of a radius $\begin{aligned} b, \end{aligned}$ as $\begin{aligned} C \end{aligned}$ rolls on the inside of a circle with center $\begin{aligned} O \end{aligned}$ and radius $\begin{aligned} a \end{aligned}$ . The initial position of $\begin{aligned} P \end{aligned}$ is $\begin{aligned} (a, 0), \end{aligned}$ and the parameter $\begin{aligned} θ \end{aligned}$ is the angle created by segments $\begin{aligned} \bar{O A} \end{aligned}$ and the diameter of circle $\begin{aligned} C \end{aligned}$ extended to meet $\begin{aligned} O \end{aligned}$ .
::我们的皮下轨道是一个曲线,由圆C的固定点P追踪,圆C半径为b,C滚动在圆的内部,圆中心为O和半径为a。P的最初位置是(a,0),而参数 __是OA段创造的角,圆C的直径延伸至O。

Step 1: Determine the parametric equations of the hypocycloid.
::第1步:确定低环流体的参数方程。

Step 2: Determine how the value of $\begin{aligned} a \end{aligned}$ impacts the graph.
::第2步:确定一个数值如何影响图形。

Step 3: Determine the impact of the graph when $\begin{aligned} b = 1 \end{aligned}$ and $\begin{aligned} a = \frac{n}{d}, \end{aligned}$ where $\begin{aligned} n \end{aligned}$ and $\begin{aligned} d \end{aligned}$ have no common factor. What happens when $\begin{aligned} n = d + 1 \end{aligned}$ ?
::第3步:确定当 b=1 和 a=n 和 a=n 没有共同系数时图形的影响。 n=d+1 会发生什么?

Step 4: Determine the impact on the graph if $\begin{aligned} a \end{aligned}$ is not rational.
::第4步:如果某一图不合理,确定该图对图的影响。

Step 5: If the circle $\begin{aligned} C \end{aligned}$ rolls outside of the circle, the curve is called an epicycloid. Determine the parametric equations for the epicycloid.
::第5步:如果圆C滚动在圆外,则曲线称为中子类。确定中子类的参数方程。

Step 6: Explain how the graph of epicycloids are impacted by changes in the radii.
::第6步:解释单子类图如何受到半径变化的影响。

Section outline

General

项目:极地坐标和参数等量