正交基和标准正交基

首先来看定义

Let S = {v₁, v₂, ... , v_k} be a set of vectors in Rⁿ, then S is called an orthogonal if 

        v_i . v_j=0

for all i not equal to j. An orthogonal set of vectors is called orthonormal if all vectors in S are unit vectors.

咳咳，翻译一下

对于 Rⁿ中的一个向量集合 S={v₁, v₂, ... , v_k} ， 如果存在

v_i . v_j = 0  (i != j)

那么它们就是一组正交基，如果它们都是单位向量，则它们还是标准正交基。

定理

1.给定一组正交基S = {v1, v2, ... , vk}，那么他们是线性相关的。

2.对于 Rⁿ中的一个正交基 S={v₁, v₂, ... , v_k},则 Rⁿ中的向量 v 的以S为基的第 i 个坐标为

v . v_j

例

求向量(5, 10) 在基 S = {(3/5, 4/5), (-4/5, 3/5)}中的坐标。

(5,10)^.(3/5,4/5) = 11 

(5,10)^.(-4/5, 3/5) = 2

[(5,10)]_S = (11,2)

给定一个空间向量，求出以这个向量为轴的正交基

解

假设给定的空间向量为

v = (a,b,c);

则与v垂直的向量必定满足

a*x + b*y + c*z = 0

这个向量可以是u = (0;-c;b) 或者 u = (-c,0,a);

已知两个向量，第三个向量利用向量叉乘就可以得出 

w = cross(u,v)

最后将u,v,w单位化，得到新的正交基。

圆柱体的射线求交

上面啰嗦到的两个地方这里都会用到。

首先还是定义圆柱体的类

 public class Cylinder : NGeometry
    {
        public Vector3 p0;
        public Vector3 p1;
        public float radius;

        public Cylinder(Vector3 _p0, Vector3 _p1, float _radius)
            : base(GeometryType.Cylinder)
        {
            p0 = _p0;
            p1 = _p1;
            radius = _radius;
        }

        public Cylinder() : base(GeometryType.Cylinder) { }

        public Vector3 ComputeDirection()
        {
            return p1 - p0;
        }
    }

看检测的代码，有点长

 public static bool Raycast(Ray ray, float distance, Cylinder cylinder, out RaycastHitInfo hitInfo)
        {
            hitInfo = new RaycastHitInfo();
            Vector3 cylinderDir = cylinder.ComputeDirection();
            Vector3 kW = cylinderDir;
            float fWLength = kW.magnitude;
            kW.Normalize();

            //two thin for check
            if (fWLength <= 1e-6f)
            {
                return false;
            }

            //generate orthonormal basis
            //cylinder along the z direction
            Vector3 kU = Vector3.zero;
            if (fWLength > 0.0f)
            {
                float fInvLength;
                if (Mathf.Abs(kW.x) >= Mathf.Abs(kW.y))
                {
                    fInvLength = 1.0f / Mathf.Sqrt(kW.x * kW.x + kW.z * kW.z);
                    kU.x = -kW.z * fInvLength;
                    kU.y = 0.0f;
                    kU.z = kW.x * fInvLength;
                }
                else
                {
                    // W.y or W.z is the largest magnitude component, swap them
                    fInvLength = 1.0f / Mathf.Sqrt(kW.y * kW.y + kW.z * kW.z);
                    kU.x = 0.0f;
                    kU.y = kW.z * fInvLength;
                    kU.z = -kW.y * fInvLength;
                }
            }
            Vector3 kV = Vector3.Cross(kW, kU);
            kV.Normalize();

            // compute intersection
            //Transform the ray to the cylinder's local coordinate
            //new Ray direction
            Vector3 kD = new Vector3(Vector3.Dot(kU, ray.direction), Vector3.Dot(kV, ray.direction), Vector3.Dot(kW, ray.direction));
            float fDLength = kD.magnitude;
            Debug.Log("fDLength: " + fDLength);
            kD.Normalize();

            float fInvDLength = 1.0f / fDLength;
            Vector3 kDiff = ray.origin - cylinder.p0;
            //new Ray origin
            Vector3 kP = new Vector3(Vector3.Dot(kU, kDiff), Vector3.Dot(kV, kDiff), Vector3.Dot(kW, kDiff));

            float fRadiusSqr = cylinder.radius * cylinder.radius;

            // Is the ray direction parallel to the cylinder direction? (or zero)
            if (Mathf.Abs(kD.z) >= 1.0f - Mathf.Epsilon || fDLength < Mathf.Epsilon)
            {
                float fAxisDir = Vector4.Dot(ray.direction, cylinderDir);

                float fDiscr = fRadiusSqr - kP.x * kP.x - kP.y * kP.y;
                // ray direction anti-parallel to the cylinder direction
                if (fAxisDir < 0 && fDiscr >= 0.0f)
                {
                    if (kP.z > fWLength)
                    {
                        hitInfo.distance = (kP.z - fWLength) * fInvDLength;
                    }
                    else if (kP.z < 0)
                    {
                        return false;
                    }
                    else if (kP.z > 0 && kP.z < fWLength)
                    {
                        hitInfo.distance = kP.z * fInvDLength;
                    }

                    if (hitInfo.distance > distance)
                        return false;
                    hitInfo.point = hitInfo.distance * ray.direction;
                    return true;
                }
                // ray direction parallel to the cylinder direction
                else if (fAxisDir > 0 && fDiscr >= 0.0f)
                {
                    if (kP.z > fWLength)
                    {
                        return false;
                    }
                    else if (kP.z < 0)
                    {
                        hitInfo.distance = -kP.z * fInvDLength;
                    }
                    else if (kP.z > 0 && kP.z < fWLength)
                    {
                        hitInfo.distance = (fWLength - kP.z) * fInvDLength;
                    }
                    if (hitInfo.distance > distance)
                        return false;
                    hitInfo.point = hitInfo.distance * ray.direction;
                    return true;
                }
                else
                {
                    //ray origin out of the circle
                    return false;
                }
            }

            // test intersection with infinite cylinder
            // set up quadratic Q(t) = a*t^2 + 2*b*t + c
            float fA = kD.x * kD.x + kD.y * kD.y;
            float fB = kP.x * kD.x + kP.y * kD.y;
            float fC = kP.x * kP.x + kP.y * kP.y - fRadiusSqr;
            float delta = fB * fB - fA * fC;
            // line does not intersect infinite cylinder
            if (delta < 0.0f)
            {
                return false;
            }

            // line intersects infinite cylinder in two points
            if (delta > 0.0f)
            {
                float fRoot = Mathf.Sqrt(delta);
                float fInv = 1.0f / fA;
                float fT = (-fB - fRoot) * fInv;
                float fTmp = kP.z + fT * kD.z;
                float dist0 = 0f, dist1 = 0f;

                float fT1 = (-fB + fRoot) * fInv;
                float fTmp1 = kP.z + fT * kD.z;

                //cast two point
                //fTmp <= fWLength to check intersect point between slab.
                if ((0.0f <= fTmp && fTmp <= fWLength) && (0.0f <= fTmp1 && fTmp1 <= fWLength))
                {
                    dist0 = fT * fInvDLength;
                    dist1 = fT1 * fInvDLength;
                    hitInfo.distance = Mathf.Min(dist0, dist1);
                    return true;
                }
                else if ((0.0f <= fTmp && fTmp <= fWLength))
                {
                    dist0 = fT * fInvDLength;
                    hitInfo.distance = dist0;
                    return true;
                }
                else if ((0.0f <= fTmp1 && fTmp1 <= fWLength))
                {
                    dist1 = fT1 * fInvDLength;
                    hitInfo.distance = dist1;
                    return true;
                }


                //If intersect the infinite cylinder but point not between slab, the ray may intersect cylinder's caps.
                //Test intersection with caps
                float deltaAngle = Vector4.Dot(ray.direction, cylinderDir);

                // Ray direction anti-parallel to the capsule direction
                if (deltaAngle < 0)
                {
                    if (kP.z > fWLength)
                    {
                        float deltaZ = kP.z - fWLength;
                        float angle = Vector3.Angle(ray.direction, -cylinderDir);
                        hitInfo.distance = (kP.z - fWLength) * fInvDLength / Mathf.Cos(angle * Mathf.Deg2Rad);
                    }
                    else if (kP.z < 0)
                    {
                        Debug.Log("No cap0");
                        return false;
                    }

                    if (hitInfo.distance > distance)
                        return false;
                    hitInfo.point = ray.origin + hitInfo.distance * ray.direction;
                    if (Vector3.Distance(hitInfo.point, cylinder.p1) > cylinder.radius)
                    {
                        return false;
                    }
                    return true;
                }
                // Direction parallel to the cylinder direction
                else if (deltaAngle > 0)
                {
                    if (kP.z > fWLength)
                    {
                        Debug.Log("No cap1");
                        return false;
                    }
                    else if (kP.z < 0)
                    {
                        float angle = Vector3.Angle(ray.direction, cylinderDir);
                        hitInfo.distance = -kP.z * fInvDLength / Mathf.Cos(angle * Mathf.Deg2Rad);
                    }
                    if (hitInfo.distance > distance)
                        return false;
                    hitInfo.point = ray.origin + hitInfo.distance * ray.direction;
                    if (Vector3.Distance(hitInfo.point, cylinder.p0) > cylinder.radius)
                    {
                        return false;
                    }
                    return true;
                }
            }
            // line is tangent to infinite cylinder
            else
            {
                float fT = -fB / fA;
                float fTmp = kP.z + fT * kD.z;
                if (0.0f <= fTmp && fTmp <= fWLength)
                {
                    hitInfo.distance = fT * fInvDLength;
                    return true;
                }
            }
            return false;
        }

简单梳理一下流程

1.将坐标系转换到了以圆柱体方向为轴的坐标系，用的就是给定一个空间向量，求出以这个向量为轴的正交基中提到的方法；

2.将Ray的origin和Direction都转到新的坐标系下，用的是上面的定理2。新的ray的origin为kP，direction为kD，射线上的点可以表示为kP+t*kD；

3.判断射线方向是否与Z轴方向平行，如果是，判断是否是圆柱体的上下面相交；

4.判断是否是圆柱相交，这里先假设圆柱是无限的，判断的方法是求解二次方程。这里详细说一下，二次函数的形式为

Q(t) = a*t^2 + 2*b*t + c

假设设想和圆柱相交，则必定存在射线上存在t，满足点kP+t*kD到Z轴的距离为圆柱的半径

(kP.x +t*kD.x)^2 +(kP.y +t*kD.y)^2 = R^2

先通过delta判断根的个数，最后求解就可以得到结果。

测试代码

using UnityEngine;
using System.Collections;
using NPhysX;

public class RayCylinderTester : MonoBehaviour {

    public GameObject cylinder;
    Cylinder _cylinder;
    Ray ray;
    float castDistance = 10f;
    // Use this for initialization
    void Start () {
        _cylinder = new Cylinder();
    }
	
	// Update is called once per frame
	void Update () {
        ray = new Ray(Vector3.zero, new Vector3(1, 1, 1));
        _cylinder.radius = 0.5f * cylinder.transform.localScale.x;
        _cylinder.p0 = cylinder.transform.position + cylinder.transform.rotation * Vector3.down * cylinder.transform.localScale.y;
        _cylinder.p1 = cylinder.transform.position + cylinder.transform.rotation * Vector3.up * cylinder.transform.localScale.y;

        Debug.DrawLine(_cylinder.p0, _cylinder.p1, Color.green);

        RaycastHitInfo hitinfo = new RaycastHitInfo();

        if (NRaycastTests.Raycast(ray, castDistance, _cylinder, out hitinfo))
        {
            Debug.DrawLine(ray.origin, ray.origin + ray.direction * hitinfo.distance, Color.red, 0, true);
        }
        else
        {
            Debug.DrawLine(ray.origin, ray.origin + ray.direction * castDistance, Color.blue, 0, true);
        }
    }
}

参考

Orthonormal Bases in Rn - http://ltcconline.net/greenl/courses/203/vectors/orthonormalbases.htm

PhysX 3.3  source code