以给定角度在矩形上查找点

时间：2010-10-31 02:28:37

标签： python math graphics geometry

我正在尝试使用给定角度（Theta）在矩形对象中绘制渐变，其中渐变的末端接触矩形的周长。

Graph

我认为使用切线会起作用，但是我很难解决问题。有一个简单的算法，我只是缺少？

最终结果

因此，这将是（angle，RectX1，RectX2，RectY1，RectY2）的函数。我希望它以[x1，x2，y1，y2]的形式返回，以便渐变将在正方形上绘制。 在我的问题中，如果原点是0，那么x2 = -x1和y2 = -y1。但它并不总是出现在原点上。

8 个答案:

答案 0 :(得分：35)

让我们调用 a 和 b 您的矩形边，然后（x0，y0）调整矩形中心的坐标。

您需要考虑四个区域：

alt text 

    Region    from               to                 Where
    ====================================================================
       1      -arctan(b/a)       +arctan(b/a)       Right green triangle
       2      +arctan(b/a)        π-arctan(b/a)     Upper yellow triangle
       3       π-arctan(b/a)      π+arctan(b/a)     Left green triangle
       4       π+arctan(b/a)     -arctan(b/a)       Lower yellow triangle

使用一点trigonometry-fu，我们可以获得每个区域中所需交点的坐标。

alt text

所以 Z0 是区域1和3的交点的表达式 并且 Z1 是区域2和4的交叉点的表达式

根据区域，所需的线从（X0，Y0）传递到Z0或Z1。所以记住Tan（φ）= Sin（φ）/ Cos（φ）


    Lines in regions      Start                   End
    ======================================================================
       1 and 3           (X0,Y0)      (X0 + a/2 , (a/2 * Tan(φ))+ Y0
       2 and 4           (X0,Y0)      (X0 + b/(2* Tan(φ)) , b/2 + Y0)

请注意每个象限中Tan（φ）的符号，并且始终从THE POSITIVE x轴ANTICLOCKWISE测量角度。

HTH！

答案 1 :(得分：10)

好的，哇！，我终于得到了这个。

注意：我的理由是基于belisarius的精彩答案。如果你喜欢这个，请也喜欢他。我所做的只是将他所说的内容转化为代码。

这是Objective-C中的样子。它应该很简单，可以转换为您喜欢的语言。

+ (CGPoint) edgeOfView: (UIView*) view atAngle: (float) theta
{
    // Move theta to range -M_PI .. M_PI
    const double twoPI = M_PI * 2.;
    while (theta < -M_PI)
    {
        theta += twoPI;
    }

    while (theta > M_PI)
    {
        theta -= twoPI;
    }

    // find edge ofview
    // Ref: http://stackoverflow.com/questions/4061576/finding-points-on-a-rectangle-at-a-given-angle
    float aa = view.bounds.size.width;                                          // "a" in the diagram
    float bb = view.bounds.size.height;                                         // "b"

    // Find our region (diagram)
    float rectAtan = atan2f(bb, aa);
    float tanTheta = tan(theta);

    int region;
    if ((theta > -rectAtan)
    &&  (theta <= rectAtan) )
    {
        region = 1;
    }
    else if ((theta >  rectAtan)
    &&       (theta <= (M_PI - rectAtan)) )
    {
        region = 2;
    }
    else if ((theta >   (M_PI - rectAtan))
    ||       (theta <= -(M_PI - rectAtan)) )
    {
        region = 3;
    }
    else
    {
        region = 4;
    }

    CGPoint edgePoint = view.center;
    float xFactor = 1;
    float yFactor = 1;

    switch (region)
    {
        case 1: yFactor = -1;       break;
        case 2: yFactor = -1;       break;
        case 3: xFactor = -1;       break;
        case 4: xFactor = -1;       break;
    }

    if ((region == 1)
    ||  (region == 3) )
    {
        edgePoint.x += xFactor * (aa / 2.);                                     // "Z0"
        edgePoint.y += yFactor * (aa / 2.) * tanTheta;
    }
    else                                                                        // region 2 or 4
    {
        edgePoint.x += xFactor * (bb / (2. * tanTheta));                        // "Z1"
        edgePoint.y += yFactor * (bb /  2.);
    }

    return edgePoint;
}

此外，这是我创建的一个小测试视图，用于验证它是否有效。创建此视图并将其放置在某个位置，它将在边缘处进行另一个小视图扫描。

@interface DebugEdgeView()
{
    int degrees;
    UIView *dotView;
    NSTimer *timer;
}

@end

@implementation DebugEdgeView

- (void) dealloc
{
    [timer invalidate];
}


- (id) initWithFrame: (CGRect) frame
{
    self = [super initWithFrame: frame];
    if (self)
    {
        self.backgroundColor = [[UIColor magentaColor] colorWithAlphaComponent: 0.25];
        degrees = 0;
        self.clipsToBounds = NO;

        // create subview dot
        CGRect dotRect = CGRectMake(frame.size.width / 2., frame.size.height / 2., 20, 20);
        dotView = [[DotView alloc] initWithFrame: dotRect];
        dotView.backgroundColor = [UIColor magentaColor];
        [self addSubview: dotView];

        // move it around our edges
        timer = [NSTimer scheduledTimerWithTimeInterval: (5. / 360.)
                                                 target: self
                                               selector: @selector(timerFired:)
                                               userInfo: nil
                                                repeats: YES];
    }

    return self;
}


- (void) timerFired: (NSTimer*) timer
{
    float radians = ++degrees * M_PI / 180.;
    if (degrees > 360)
    {
        degrees -= 360;
    }

    dispatch_async(dispatch_get_main_queue(), ^{
        CGPoint edgePoint = [MFUtils edgeOfView: self atAngle: radians];
        edgePoint.x += (self.bounds.size.width  / 2.) - self.center.x;
        edgePoint.y += (self.bounds.size.height / 2.) - self.center.y;
        dotView.center = edgePoint;
    });
}

@end

答案 2 :(得分：6)

Javascript版本：

＆＃13;
＆＃13; 
function edgeOfView(rect, deg) {
  var twoPI = Math.PI*2;
  var theta = deg * Math.PI / 180;
  
  while (theta < -Math.PI) {
    theta += twoPI;
  }
  
  while (theta > Math.PI) {
    theta -= twoPI;
  }
  
  var rectAtan = Math.atan2(rect.height, rect.width);
  var tanTheta = Math.tan(theta);
  var region;
  
  if ((theta > -rectAtan) && (theta <= rectAtan)) {
      region = 1;
  } else if ((theta > rectAtan) && (theta <= (Math.PI - rectAtan))) {
      region = 2;
  } else if ((theta > (Math.PI - rectAtan)) || (theta <= -(Math.PI - rectAtan))) {
      region = 3;
  } else {
      region = 4;
  }
  
  var edgePoint = {x: rect.width/2, y: rect.height/2};
  var xFactor = 1;
  var yFactor = 1;
  
  switch (region) {
    case 1: yFactor = -1; break;
    case 2: yFactor = -1; break;
    case 3: xFactor = -1; break;
    case 4: xFactor = -1; break;
  }
  
  if ((region === 1) || (region === 3)) {
    edgePoint.x += xFactor * (rect.width / 2.);                                     // "Z0"
    edgePoint.y += yFactor * (rect.width / 2.) * tanTheta;
  } else {
    edgePoint.x += xFactor * (rect.height / (2. * tanTheta));                        // "Z1"
    edgePoint.y += yFactor * (rect.height /  2.);
  }
  
  return edgePoint;
};
＆＃13;
＆＃13;
＆＃13;

答案 3 :(得分：3)

在你的照片之后，我将假设矩形以（0,0）为中心，右上角为（w，h）。然后连接（0,0）到（w，h）的线与X轴形成角度φ，其中tan（φ）= h / w。

假设θ> φ，我们正在寻找点（x，y），其中您绘制的线与矩形的上边缘相交。然后y / x = tan（θ）。我们知道y = h所以，求解x，得到x = h / tan（θ）。

如果θ< φ，该线与（x，y）处矩形的右边缘相交。这一次，我们知道x = w，所以y = tan（θ）* w。

答案 4 :(得分：1)

Find the CGPoint on a UIView rectangle intersected by a straight line at a given angle from the center point涉及以下步骤，对这个问题有一个很好的（更多程序化的iOS / Objective-C）答案：

  1. 假设角度大于或等于0且小于2 *π，从0（东）逆时针方向前进。
  2. 获取与矩形右边缘的交点的y坐标[tan（angle）* width / 2]。
  3. 检查此y坐标是否在矩形框内（绝对值小于或等于高度的一半）。
  4. 如果y交点在矩形中，则如果角度小于π/ 2或大于3π/ 2，则选择右边缘（宽度/ 2，-y坐标）。否则选择左边缘（-width / 2，y coord）。
  5. 如果右边缘交点的y坐标超出界限，则计算与底边相交的x坐标[高度的一半/ tan（角度）]。
  6. 接下来确定您是想要上边缘还是下边缘。如果角度小于π，我们需要底边（x， - 半高）。否则，我们需要顶边（-x coord，高度的一半）。
  7. 然后（如果帧的中心不是0,0），将该点偏移到帧的实际中心。

答案 5 :(得分：0)

对于Java，是LibGDX。我将角度设为两倍以提高精度。

public static Vector2 projectToRectEdge(double angle, float width, float height, Vector2 out)
{
    return projectToRectEdgeRad(Math.toRadians(angle), width, height, out);
}

public static Vector2 projectToRectEdgeRad(double angle, float width, float height, Vector2 out)
{
    float theta = negMod((float)angle + MathUtils.PI, MathUtils.PI2) - MathUtils.PI;

    float diag = MathUtils.atan2(height, width);
    float tangent = (float)Math.tan(angle);

    if (theta > -diag && theta <= diag)
    {
        out.x = width / 2f;
        out.y = width / 2f * tangent;
    }
    else if(theta > diag && theta <= MathUtils.PI - diag)
    {
        out.x = height / 2f / tangent;
        out.y = height / 2f;
    }
    else if(theta > MathUtils.PI - diag && theta <= MathUtils.PI + diag)
    {
        out.x = -width / 2f;
        out.y = -width / 2f * tangent;
    }
    else
    {
        out.x = -height / 2f / tangent;
        out.y = -height / 2f;
    }

    return out;
}

答案 6 :(得分：0)

虚幻引擎4（UE4）C ++版本。

注意：这基于Olie的Code。基于Belisarius的Answer。如果这可以帮助您，请给这些人投票。

更改：使用UE4语法和功能，并且取反了Angle。

标题

UFUNCTION(BlueprintCallable, meta = (DisplayName = "Project To Rectangle Edge (Radians)"), Category = "Math|Geometry")
static void ProjectToRectangleEdgeRadians(FVector2D Extents, float Angle, FVector2D & EdgeLocation);

代码

void UFunctionLibrary::ProjectToRectangleEdgeRadians(FVector2D Extents, float Angle, FVector2D & EdgeLocation)
{
    // Move theta to range -M_PI .. M_PI. Also negate the angle to work as expected.
    float theta = FMath::UnwindRadians(-Angle);

    // Ref: http://stackoverflow.com/questions/4061576/finding-points-on-a-rectangle-at-a-given-angle
    float a = Extents.X; // "a" in the diagram | Width
    float b = Extents.Y; // "b"                | Height

    // Find our region (diagram)
    float rectAtan = FMath::Atan2(b, a);
    float tanTheta = FMath::Tan(theta);

    int region;
    if ((theta > -rectAtan) && (theta <= rectAtan))
    {
        region = 1;
    }
    else if ((theta > rectAtan) && (theta <= (PI - rectAtan)))
    {
        region = 2;
    }
    else if ((theta > (PI - rectAtan)) || (theta <= -(PI - rectAtan)))
    {
        region = 3;
    }
    else
    {
        region = 4;
    }

    float xFactor = 1.f;
    float yFactor = 1.f;

    switch (region)
    {
        case 1: yFactor = -1; break;
        case 2: yFactor = -1; break;
        case 3: xFactor = -1; break;
        case 4: xFactor = -1; break;
    }

    EdgeLocation = FVector2D(0.f, 0.f); // This rese is nessesary, UE might re-use otherwise. 

    if (region == 1 || region == 3)
    {
        EdgeLocation.X += xFactor * (a / 2.f);              // "Z0"
        EdgeLocation.Y += yFactor * (a / 2.f) * tanTheta;
    }
    else // region 2 or 4
    {
        EdgeLocation.X += xFactor * (b / (2.f * tanTheta)); // "Z1"
        EdgeLocation.Y += yFactor * (b / 2.f);
    }
}

答案 7 :(得分：0)

蟒蛇

import math
import matplotlib.pyplot as plt

twoPI = math.pi * 2.0
PI = math.pi

def get_points(width, height, theta):
    theta %= twoPI

    aa = width
    bb = height

    rectAtan = math.atan2(bb,aa)
    tanTheta = math.tan(theta)

    xFactor = 1
    yFactor = 1
    
    # determine regions
    if theta > twoPI-rectAtan or theta <= rectAtan:
        region = 1
    elif theta > rectAtan and theta <= PI-rectAtan:
        region = 2

    elif theta > PI - rectAtan and theta <= PI + rectAtan:
        region = 3
        xFactor = -1
        yFactor = -1
    elif theta > PI + rectAtan and theta < twoPI - rectAtan:
        region = 4
        xFactor = -1
        yFactor = -1
    else:
        print(f"region assign failed : {theta}")
        raise
    
    # print(region, xFactor, yFactor)
    edgePoint = [0,0]
    ## calculate points
    if (region == 1) or (region == 3):
        edgePoint[0] += xFactor * (aa / 2.)
        edgePoint[1] += yFactor * (aa / 2.) * tanTheta
    else:
        edgePoint[0] += xFactor * (bb / (2. * tanTheta))
        edgePoint[1] += yFactor * (bb /  2.)

    return region, edgePoint

l_x = []
l_y = []
theta = 0
for _ in range(10000):
    r, (x, y) = get_points(600,300, theta)
    l_x.append(x)
    l_y.append(y)
    theta += (0.01 / PI)

    if _ % 100 == 0:
        print(r, x,y)

plt.plot(l_x, l_y)
plt.show()
相关问题
最新问题