diff --git a/utils/geometry/geometry.go b/utils/geometry/geometry.go
index 30d7f45..0d24c69 100644
--- a/utils/geometry/geometry.go
+++ b/utils/geometry/geometry.go
@@ -139,3 +139,14 @@ func CalculateNewCoordinate[V generic.SignedNumber](x, y, angle, distance V) (ne
 
 	return newX, newY
 }
+
+// CalcAngleDifference 计算两个角度之间的最小角度差
+func CalcAngleDifference[V generic.Number](angleA, angleB V) V {
+	pi := math.Pi
+	t := angleA - angleB
+	a := t + V(pi)
+	b := V(pi) * 2
+	t = V(math.Floor(float64(a/b))) * b
+	t -= pi
+	return t
+}
diff --git a/utils/geometry/line.go b/utils/geometry/line.go
index b4c6a6f..62bfcee 100644
--- a/utils/geometry/line.go
+++ b/utils/geometry/line.go
@@ -2,8 +2,41 @@ package geometry
 
 import (
 	"github.com/kercylan98/minotaur/utils/generic"
+	"github.com/kercylan98/minotaur/utils/maths"
+	"sort"
 )
 
+// NewLine 创建一根线段
+func NewLine[V generic.SignedNumber](start, end Point[V]) Line[V] {
+	if start.Equal(end) {
+		panic("two points of the line segment are the same")
+	}
+	return Line[V]{start, end}
+}
+
+// Line 通过两个点表示一根线段
+type Line[V generic.SignedNumber] [2]Point[V]
+
+// GetPoints 获取该线段的两个点
+func (slf Line[V]) GetPoints() [2]Point[V] {
+	return slf
+}
+
+// GetStart 获取该线段的开始位置
+func (slf Line[V]) GetStart() Point[V] {
+	return slf[0]
+}
+
+// GetEnd 获取该线段的结束位置
+func (slf Line[V]) GetEnd() Point[V] {
+	return slf[1]
+}
+
+// GetLength 获取该线段的长度
+func (slf Line[V]) GetLength() V {
+	return CalcDistance(DoublePointToCoordinate(slf.GetStart(), slf.GetEnd()))
+}
+
 // PointOnLineWithCoordinate 通过一个线段两个点的位置和一个点的坐标，判断这个点是否在一条线段上
 func PointOnLineWithCoordinate[V generic.SignedNumber](x1, y1, x2, y2, x, y V) bool {
 	return (x-x1)*(y2-y1) == (x2-x1)*(y-y1)
@@ -48,3 +81,38 @@ func PointOnSegmentWithCoordinateArray[V generic.SignedNumber](point1, point2, p
 	x, y := point.GetXY()
 	return x >= x1 && x <= x2 && y >= y1 && y <= y2 && PointOnLineWithCoordinate(x1, y1, x2, y2, x, y)
 }
+
+// CalcLineIsCollinear 检查两条线段在一个误差内是否共线
+//   - 共线是指两条线段在同一直线上，即它们的延长线可以重合
+func CalcLineIsCollinear[V generic.SignedNumber](line1, line2 Line[V], tolerance V) bool {
+	area1 := CalcTriangleTwiceArea(line1.GetStart(), line1.GetEnd(), line2.GetStart())
+	area2 := CalcTriangleTwiceArea(line1.GetStart(), line1.GetEnd(), line2.GetEnd())
+	return maths.Tolerance(area1, 0, tolerance) && maths.Tolerance(area2, 0, tolerance)
+}
+
+// CalcLineIsOverlap 通过对点进行排序来检查两条共线线段是否重叠，返回重叠线段
+func CalcLineIsOverlap[V generic.SignedNumber](line1, line2 Line[V]) (line Line[V], overlap bool) {
+	var shapes = []Shape[V]{
+		{line1.GetStart(), line1.GetEnd(), line1.GetStart()},
+		{line1.GetStart(), line1.GetEnd(), line1.GetEnd()},
+		{line2.GetStart(), line2.GetEnd(), line2.GetStart()},
+		{line2.GetStart(), line2.GetEnd(), line2.GetEnd()},
+	}
+	sort.Slice(shapes, func(i, j int) bool {
+		a, b := shapes[i], shapes[j]
+		if a[2].GetX() < b[2].GetX() {
+			return true
+		} else if a[2].GetX() > b[2].GetX() {
+			return false
+		} else {
+			return a[2].GetY() < b[2].GetY()
+		}
+	})
+
+	notOverlap := shapes[0][0].Equal(shapes[1][0]) && shapes[0][1].Equal(shapes[1][1])
+	singlePointOverlap := shapes[1][2].Equal(shapes[2][2])
+	if notOverlap || singlePointOverlap {
+		return line, false
+	}
+	return NewLine(shapes[1][2], shapes[2][2]), true
+}
diff --git a/utils/geometry/rectangle.go b/utils/geometry/rectangle.go
index 58fc7fb..cef4ff1 100644
--- a/utils/geometry/rectangle.go
+++ b/utils/geometry/rectangle.go
@@ -322,3 +322,14 @@ func GetRectangleFullPos[V generic.SignedNumber](width, height V) (result []V) {
 	}
 	return
 }
+
+// CalcRectangleCentroid 计算矩形质心
+//   - 非多边形质心计算，仅为顶点的平均值 - 该区域中多边形因子的适当质心
+func CalcRectangleCentroid[V generic.SignedNumber](shape Shape[V]) Point[V] {
+	x, y := V(0), V(0)
+	length := V(shape.PointCount())
+
+	x /= length
+	y /= length
+	return NewPoint(x, y)
+}
diff --git a/utils/geometry/shape.go b/utils/geometry/shape.go
index f7f17e3..f9c340a 100644
--- a/utils/geometry/shape.go
+++ b/utils/geometry/shape.go
@@ -11,9 +11,19 @@ import (
 )
 
 var (
-	ShapeStringHasBorder = false // 控制 Shape.String 是否拥有边界
+	shapeStringHasBorder = false // 控制 Shape.String 是否拥有边界
 )
 
+// SetShapeStringHasBorder 设置 Shape.String 是拥有边界的
+func SetShapeStringHasBorder() {
+	shapeStringHasBorder = true
+}
+
+// SetShapeStringNotHasBorder 设置 Shape.String 是没有边界的
+func SetShapeStringNotHasBorder() {
+	shapeStringHasBorder = false
+}
+
 // NewShape 通过多个点生成一个形状进行返回
 func NewShape[V generic.SignedNumber](points ...Point[V]) Shape[V] {
 	return points
@@ -63,7 +73,7 @@ func (slf Shape[V]) String() string {
 	left, right, top, bottom := GetShapeCoverageAreaWithCoordinateArray(slf.Points()...)
 	width := right - left + 1
 	height := bottom - top + 1
-	if !ShapeStringHasBorder {
+	if !shapeStringHasBorder {
 		for y := top; y < top+height; y++ {
 			for x := left; x < left+width; x++ {
 				exist := false
@@ -334,6 +344,20 @@ func (slf Shape[V]) getAllGraphicComposition(opt *shapeSearchOptions) (result []
 	return result
 }
 
+// Edges 获取该形状每一条边
+//   - 该形状需要最少由3个点组成，否则将不会返回任意一边
+func (slf Shape[V]) Edges() (edges []Line[V]) {
+	if len(slf) < 3 {
+		return
+	}
+	for i := 1; i < slf.PointCount(); i++ {
+		before := slf[i-1]
+		edges = append(edges, NewLine(before, slf[i]))
+	}
+	edges = append(edges, NewLine(slf[0], slf[len(slf)-1]))
+	return edges
+}
+
 // getAllGraphicCompositionWithAsc 通过升序的方式获取该形状中包含的所有图形组合及其位置
 //   - 升序指标为图形包含的点数量
 //   - 其余内容可参考 getAllGraphicComposition
@@ -355,3 +379,37 @@ func (slf Shape[V]) getAllGraphicCompositionWithDesc(opt *shapeSearchOptions) (r
 	})
 	return
 }
+
+// CalcBoundingRadius 计算多边形转换为圆的半径
+func CalcBoundingRadius[V generic.SignedNumber](shape Shape[V]) V {
+	var boundingRadius V
+	var centroid = CalcRectangleCentroid(shape)
+	for _, point := range shape.Points() {
+		distance := CalcDistance(DoublePointToCoordinate(centroid, point))
+		if distance > boundingRadius {
+			boundingRadius = distance
+		}
+	}
+	return boundingRadius
+}
+
+// CalcBoundingRadiusWithCentroid 计算多边形在特定质心下圆的半径
+func CalcBoundingRadiusWithCentroid[V generic.SignedNumber](shape Shape[V], centroid Point[V]) V {
+	var boundingRadius V
+	for _, point := range shape.Points() {
+		distance := CalcDistance(DoublePointToCoordinate(centroid, point))
+		if distance > boundingRadius {
+			boundingRadius = distance
+		}
+	}
+	return boundingRadius
+}
+
+// CalcTriangleTwiceArea 计算由 a、b、c 三个点组成的三角形的面积的两倍
+func CalcTriangleTwiceArea[V generic.SignedNumber](a, b, c Point[V]) V {
+	ax := b.GetX() - a.GetX()
+	ay := b.GetY() - a.GetY()
+	bx := c.GetX() - a.GetX()
+	by := c.GetY() - a.GetY()
+	return bx*ay - ax*by
+}
diff --git a/utils/maths/math.go b/utils/maths/math.go
index 0117adc..9864792 100644
--- a/utils/maths/math.go
+++ b/utils/maths/math.go
@@ -2,8 +2,18 @@ package maths
 
 import (
 	"github.com/kercylan98/minotaur/utils/generic"
+	"math"
 )
 
+const (
+	DefaultTolerance = 0.0001 // 默认误差范围
+)
+
+// GetDefaultTolerance 获取默认误差范围
+func GetDefaultTolerance() float64 {
+	return DefaultTolerance
+}
+
 // Pow 整数幂运算
 func Pow(a, n int) int {
 	if a == 0 {
@@ -94,3 +104,8 @@ func Clamp[V generic.Number](value, min, max V) V {
 	}
 	return value
 }
+
+// Tolerance 检查两个值是否在一个误差范围内
+func Tolerance[V generic.Number](value1, value2, tolerance V) bool {
+	return V(math.Abs(float64(value1-value2))) <= tolerance
+}