三维中圆的参数化表达

生成圆环的时候我们经常需要通过扫掠生成一个离散的平面圆路径点集。平面圆的数学表示定义考虑n维空间的圆中心点C,圆上点P半径r平面法向为N其满足{∣P⃗−C⃗∣r(P⃗−C⃗)⋅N⃗0 \begin{cases} |\vec{P} - \vec{C}| r \\ (\vec{P} - \vec{C}) \cdot \vec{N} 0 \end{cases}{∣P−C∣r(P−C)⋅N0​推导考虑二维圆的平面方程为(x−a)2(y−b)2r2(x-a)^2(y-b)^2r^2(x−a)2(y−b)2r2其参数化方程为{x(θ)arcos⁡θy(θ)brsin⁡θ,θ∈[0,2π) \begin{cases} x(\theta) a r\cos\theta \\ y(\theta) b r\sin\theta \end{cases}, \theta \in [0, 2\pi){x(θ)arcosθy(θ)brsinθ​,θ∈[0,2π)考虑三维上面的例子其本质是圆平面坐标系通过换基矩阵到达了整体的坐标系如图考虑圆所在平面的法向量N⃗\vec{N}N归一化后记为k^\hat{k}k^在平面内取两个互相垂直的单位向量i^\hat{i}i^、j^\hat{j}j^​构成换基矩阵[i^, j^, k^][\hat{i},\ \hat{j},\ \hat{k}][i^,j^​,k^]其中i^,j^,k^\hat{i},\hat{j},\hat{k}i^,j^​,k^均为在全局坐标系下的表达。则圆的三维参数化方程为P⃗(θ)C⃗rcos⁡θ i^rsin⁡θ j^,θ∈[0,2π) \vec{P}(\theta) \vec{C} r\cos\theta\,\hat{i} r\sin\theta\,\hat{j}, \quad \theta \in [0, 2\pi)P(θ)Crcosθi^rsinθj^​,θ∈[0,2π)其矩阵表达为P⃗(θ)C⃗[i^, j^, k^][rcos⁡θrsin⁡θ0],θ∈[0,2π) \vec{P}(\theta) \vec{C} [\hat{i},\ \hat{j},\ \hat{k}] \begin{bmatrix} r\cos\theta \\ r\sin\theta \\ 0\end{bmatrix}, \quad \theta \in [0, 2\pi)P(θ)C[i^,j^​,k^]​rcosθrsinθ0​​,θ∈[0,2π)代码/** 单位圆离散点: p(i) u·cos(i/n·2π) v·sin(i/n·2π) */functiondiscreteRing(u,v,segments){constpositions[];for(leti0;isegments;i1){constt(i/segments)*Cesium.Math.TWO_PI;constcMath.cos(t);constsMath.sin(t);positions.push(newCesium.Cartesian3(u.x*cv.x*s,u.y*cv.y*s,u.z*cv.z*s,),);}returnpositions;}functiontoWorld(localUnit,radius){returnlocalUnit.map((p){constscaledCesium.Cartesian3.multiplyByScalar(p,radius,newCesium.Cartesian3());returnCesium.Matrix4.multiplyByPoint(frame,scaled,newCesium.Cartesian3());});}functionaddRing(geometry,color){viewer.scene.primitives.add(newCesium.Primitive({geometryInstances:newCesium.GeometryInstance({geometry}),appearance:newCesium.PolylineMaterialAppearance({material:Cesium.Material.fromType(Color,{color}),translucent:false,renderState:{depthTest:{enabled:false}},}),asynchronous:false,allowPicking:false,}),);}functionpolylineGeometry(positions){returnnewCesium.PolylineGeometry({positions,width:4,arcType:Cesium.ArcType.NONE,vertexFormat:Cesium.PolylineMaterialAppearance.VERTEX_FORMAT,});}functiondrawScaleAxis(){}functiondrawRotationAxis(){// Roll 绕 X(红): 圆在 YZconstgeometryXpolylineGeometry(toWorld(discreteRing(Cesium.Cartesian3.UNIT_Y,Cesium.Cartesian3.UNIT_Z,RING_SEGMENTS),RING_RADIUS),);// Pitch 绕 Y(绿): 圆在 ZXconstgeometryYpolylineGeometry(toWorld(discreteRing(Cesium.Cartesian3.UNIT_Z,Cesium.Cartesian3.UNIT_X,RING_SEGMENTS),RING_RADIUS),);// Yaw 绕 Z(蓝): 圆在 XYconstgeometryZpolylineGeometry(toWorld(discreteRing(Cesium.Cartesian3.UNIT_X,Cesium.Cartesian3.UNIT_Y,RING_SEGMENTS),RING_RADIUS),);addRing(geometryX,Cesium.Color.fromCssColorString(#e8514f));addRing(geometryY,Cesium.Color.fromCssColorString(#7ac943));addRing(geometryZ,Cesium.Color.fromCssColorString(#4a90e2));}注:本代码平面圆默认在(0,0,0)处