中点画椭圆算法_中点圆算法

以你之姓@ 2023-03-05 09:55 13阅读 0赞

中点画椭圆算法

## 中点圆算法 **(**Midpoint circle Algorithm**)** ##

This is an algorithm which is used to calculate the entire perimeter points of a circle in a first octant so that the points of the other octant can be taken easily as they are mirror points; this is due to circle property as it is symmetric about its center.

这是一种算法，用于计算第一个八分圆中一个圆的整个周边点，以便可以轻松地将另一个八分圆的点视为镜像点； 这是由于圆的属性有关它的中心对称。

![Midpoint circle algo][]

In this algorithm decision parameter is based on a circle equation. As we know that the equation of a circle is x2 \+y2 =r2 when the centre is (0, 0).

在该算法中，决策参数基于圆方程。 众所周知，当中心为(0，0)时，圆的方程为x 2 \+ y 2 = r 2 。

Now let us define the function of a circle i.e.: **fcircle(x,y)= x2 \+y2 \- r2**

现在让我们定义一个圆的函数，即： **fcircle(x，y)= x 2 \+ y 2 \-r 2**

1.  If **fcircle < 0** then **x**, **y** is inside the circle boundary.
    
    如果**fcircle <0，**则**x** ， **y**在圆边界之内。
2.  If **fcircle > 0** then **x**, **y** is outside the circle boundary.
    
    如果**fcircle> 0，**则**x** ， **y**在圆边界之外。
3.  If **fcircle = 0** then **x**, **y** is on the circle boundary.
    
    如果**fcircle = 0，**则**x** ， **y**在圆边界上。

### 决策参数 **(**Decision parameter**)** ###

**pk =fcircle(xk+1,yk-1/2)** where **pk** is a decision parameter and in this **½** is taken because it is a midpoint value through which it is easy to calculate value of **yk** and **yk-1**.

**p k = fcircle(x k + 1 ，y k-1 / 2 )** ，其中**p k**是决策参数，在此**1/2**中采用**p k**是因为它是一个中点值，通过该中点值很容易计算**y k**和**y k -1** 。

I.e. **pk= (xk+1)2\+ (yk-1/2)2\-r2**

即**p k =(x k + 1 ) 2 \+(y k-1 / 2 ) 2 \-r 2**

If **pk <0** then midpoint is inside the circle in this condition we select **y** is **yk** otherwise we will select next **y** as **yk-1** for the condition of **pk > 0**.

如果**p k <0，**则在这种情况下中点在圆内，我们选择**y**为**y k，**否则对于**p k > 0**的情况，我们将下一个**y**选择为**y k-1** 。

### 结论 **(**Conclusion**)** ###

1.  If **pk < 0** then **yk+1=yk**, by this the plotting points will be **( xk+1 ,yk)**. By this the value for the next point will be given as:
    
    如果**p k <0，**则**y k + 1 = y k** ，由此绘制点将为**(x k + 1 ，y k )** 。 这样，下一点的值将为：
    
    **Pk+1=pk \+2(xk+1) +1**
    
    **P k + 1 = p k \+2(x k + 1 )+1**
2.  If **pk > 0** then **yk+1=yk-1**, by this the plotting points will be **(xk+1, yk-1)**. By this the value of the next point will be given as:
    
    如果**p k > 0，**则**y k + 1 = y k-1** ，由此绘制点将为**(x k + 1 ，y k-1 )** 。 这样，下一点的值将为：
    
    **Pk+1=pk\+2(xk+1) +1-2(yk+1)**
    
    **P k + 1 = p k \+2(x k + 1 )+ 1-2(y k + 1 )**

### 初始决策参数 **(**Initial decision parameter**)** ###

**P0 = fcircle (1, r-1/2)**

**P 0 =圆(1，r-1 / 2)**

This is taken because of **(x0, y0) = (0, r)**

这是因为**(x 0 ，y 0 )=(0，r)**

i.e. **p0 =5/4-r or 1-r**, (**1-r** will be taken if **r** is integer)

即**p 0 = 5 / 4-r或1-r** ，(如果**r**为整数则采用**1-r** )

### 算法 **(**ALGORITHM**)** ###

1.  In this the input radius **r** is there with a centre **(xc , yc)**. To obtain the first point **m** the circumference of a circle is centered on the origin as **(x0,y0) = (0,r)**.
    
    在此，输入半径**r**以一个中心**(x c ，y c )**为中心。 为了获得第一个点**m** ，圆的圆周以**(x 0 ，y 0 )=(0，r)为中心** 。
2.  Calculate the initial decision parameters which are:
    
    计算初始决策参数为：
    
    **p0 =5/4-r or 1-r** 
    
    **p 0 = 5 / 4-r或1-r**
3.  Now at each **xk** position starting **k=0**, perform the following task.
    
    现在，在从**k = 0**开始的每个**x k**位置，执行以下任务。
    
    if
    
    如果
    
    **pk < 0** then plotting point will be **( xk+1 ,yk)** and
    
    **p k <0，**则绘图点将为**(x k + 1 ，y k )**并且
    
    **Pk+1=pk \+2(xk+1) +1**
    
    **P k + 1 = p k \+2(x k + 1 )+1**
    
    else the next point along the circle is (x
    
    否则沿圆的下一个点是\[x
    
    k+1, yk-1) and
    
    k + 1 ，y k-1 )和
    
    **Pk+1=pk\+2(xk+1) +1-2(yk+1)**
    
    **P k + 1 = p k \+2(x k + 1 )+ 1-2(y k + 1 )**
4.  Determine the symmetry points in the other quadrants.
    
    确定其他象限中的对称点。
5.  Now move at each point by the given centre that is:
    
    现在按照给定的中心在每个点处移动：
    
    **x=x+xc**
    
    **x = x + x c**
    
    **y=y+yc**
    
    **y = y + y c**
6.  At last repeat steps from 3 to 5 until the condition **x>=y**.
    
    最后重复步骤3到5，直到条件**x> = y**为止。

> 翻译自: [https://www.includehelp.com/algorithms/midpoint-circle.aspx][https_www.includehelp.com_algorithms_midpoint-circle.aspx]

中点画椭圆算法

[Midpoint circle algo]: /images/20230208/0c349d4cb6224249aaf0389aa101ee2c.png
[https_www.includehelp.com_algorithms_midpoint-circle.aspx]: https://www.includehelp.com/algorithms/midpoint-circle.aspx