Determining whether a point inside the triangle

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Determining whether a point inside the triangle

topic
Solution 1: Area Method

Algorithm thinking
Corresponding code
defect

Solution 2: Vector

Algorithm thinking
Corresponding code

Determining whether a point inside the triangle

topic

In the two-dimensional coordinate system, all values are double type, a triangle may be represented by three points,
given three points represented by triangles, give Given a point (x, Y), is determined (x, y) is in a triangle.

Solution 1: Area Method

Algorithm thinking

If a point inside the triangle ABC O, shown in Figure 9-1, then the area has an area ABC = area ABO + BCO + CAO area;
if outside the triangle ABC point O, shown in Figure 9-2, then, it has an area ABC <ABO + area area area BCO + CAO.
Triangle inside and outside 侧点

The Heron's formula , using the triangle side length determined area thereof .

Corresponding code

# 解法1：面积法
def getSideLength(x1, y1, x2, y2):
    return ((x1 - x2)**2 + (y1 - y2)**2)**0.5

def getArea(x1, y1, x2, y2, x3, y3):
    # 海伦公式：三角形边长求面积
    a = getSideLength(x1, y1, x2, y2)
    b = getSideLength(x1, y1, x3, y3)
    c = getSideLength(x2, y2, x3, y3)
    p = (a + b + c) / 2
    return (p * (p - a) * (p - b) * (p - c))  ** 0.5

def isInTri1(x1, y1, x2, y2, x3, y3, x, y):
    s = getArea(x1, y1, x2, y2, x3, y3)
    s1 = getArea(x1, y1, x2, y2, x, y)
    s2 = getArea(x1, y1, x3, y3, x, y)
    s3 = getArea(x2, y2, x3, y3, x, y)
    eps = 1e-6  # eps抵消浮点数计算过程的误差
    # 面积相等，有可能在三角形边上，因此需要排除
    if s1 < eps and s2 < eps and s3 < eps and s1 + s2 + s3 - s < eps:
        return True
    else:
        return False

defect

Prescribing other floating-point operations , the accuracy of the results led to deviations.
We need to specify eps e.g. 1e-6mitigation area aligned with the situation.

Solution 2: Vector

Algorithm thinking

If the point O in the triangle ABC, then if a little from the triangle counterclockwise through the process all sides, the point O always traveled in the left side. If the outside point O in the triangle ABC, this relationship is not satisfied. As shown below.

Using the vector product (cross product) determining the relative orientation of ( inside and outside ).

Note: maintaining the relative position of three points in the counterclockwise direction .

First need to $\ Begin {AB}$ with $\ Vec {CA}$ Cross product operation, its relative position is determined, if the result is greater than 0, then the switch B, C position.

Corresponding code

# 解法2：叉积法
def crossProduct(x1, y1, x2, y2):
    return x1 * y2 - x2 * y1

def isInTri2(x1, y1, x2, y2, x3, y3, x, y):
    # 保证三个点相对位置为逆时针方向：1 -> 2 -> 3
    if crossProduct(x2 - x1, y2 - y1, x3 - x1, y3 - y1) >= 0:
        x2, x3 = x3, x2
        y2, y3 = y3, y2
    if crossProduct(x2 - x1, y2 - y1, x - x1, y - y1) <= 0:
        return False
    if crossProduct(x3 - x2, y3 - y2, x - x2, y - y2) <= 0:
        return False
    if crossProduct(x1 - x3, y1 - y3, x - x3, y - y3) <= 0:
        return False
    return True