《Unity虚拟现实技术及其应用》 课件 vector3-matrix-quaternion

Unity的坐标系Unity采用左手坐标系Vector3Vector3类描述三维矢量和三维空间点，是三维运算中最基本的数学操作.Vector3v=newVector3(1.0f,1.0f,1.0f);print(v);Vector3的静态性质PropertiesVector3back=Vector3.back;//(0,0,-1)Vector3down=Vector3.down;//(0,-1,0)Vector3forward=Vector3.forward;//(0,0,1)Vector3left=Vector3.left;//(-1,0,0)Vector3one=Vector3.one;//(1,1,1)Vector3right=Vector3.right;//(1,0,0)Vector3up=Vector3.up;//(0,1,0)Vector3zero=Vector3.zero;//(0,0,0)Vector3的一般性质Properties获取矢量v的三个坐标分量：获取矢量v的长度：获取矢量v的单位矢量：Vector3的静态方法publicstaticfloatAngle(Vector3from,Vector3to);获取小于180度的夹角。publicstaticfloatSignedAngle(Vector3f,Vector3t,Vector3axis);获取夹角，该夹角可以大于180度。publicstaticVector3Cross(Vector3lhs,Vector3rhs);publicstaticfloatDot(Vector3lhs,Vector3rhs);Vector3a=newVector3(2,0,0);Vector3b=newVector3(0,2,0);floatx=Vector3.Dot(a,b);Vector3MoveTowards(Vector3current,Vector3target,floatmaxDistanceDelta);voidOrthoNormalize(refVector3normal,refVector3tangent);将矢量normal和tangent归一，同时将tangent方向垂直于normal。voidOrthoNormalize(refVector3normal,refVector3tangent,refVector3binormal);

Vector3

Project(Vector3

vector,

Vector3

onNormal);Projects

vector

ontovectoronNormal.Vector3ProjectOnPlane(Vector3vector,Vector3planeNormal);Projectsavectorontoaplanedefinedbyanormalorthogonaltotheplane.planeNormalVector3

Reflect(Vector3

inDirection,

Vector3

inNormal)Reflectsavectorofftheplanedefinedbyanormal.Vector3RotateTowards(Vector3current,Vector3

target,floatmaxRadiansDelta,floatmaxMagnitudeDelta);Rotatesavectorcurrenttowardstarget.staticVector3Scale(Vector3a,Vector3b);publicvoidScale(Vector3scale);Result=(a.x*b.x,a.y*b.y,a.z*b.z);

staticfloatSignedAngle(Vector3from,Vector3to,Vector3axis);Returnsthesignedangleindegreesbetweenfromandto.staticVector3Slerp(Vector3a,Vector3b,floatt);Sphericallyinterpolatesbetweentwovectors.Interpolatesbetweenaandbbyamountt.Vector3v1=newVector3(2.0f,1.0f,0.0f);Vector3v2=newVector3(-2.0f,1.0f,0.0f);for(inti=1;i<10;++i){Vector3v=Vector3.Slerp(v1,v2,0.1f*i);Debug.DrawLine(Vector3.zero,v,Color.yellow);}

staticVector3SlerpUnclamped(Vector3a,Vector3b,floatt);tcanbelessthanzeroorgreaterthanonestaticVector3SmoothDamp(Vector3current,Vector3target,refVector3currentVelocity,floatsmoothTime,floatmaxSpeed=Mathf.Infinity,floatdeltaTime=Time.deltaTime);Operatorsoperator-Subtractsonevectorfromanother.operator!=Returnstrueifvectorsdifferent.operator*Multipliesavectorbyanumber.operator/Dividesavectorbyanumber.operator+Addstwovectors.operator==Returnstrueiftwovectorsareapproximatelyequal.Vector2andVector4ScalingTransformsRotationTransformsHomogeneousCoordinatesA3DpointPisextendedtofourdimensionsbysettingitsfourthcoordinate,whichwecallthewcoordinate,equalto1.(x,y,z,w),w=1,Translationmatrix????Matrix4x4Atransformationmatrixcanperformarbitrarylinear3Dtransformations(i.e.translation,rotation,scale,shearetc.)andperspectivetransformationsusinghomogenouscoordinates.先平移，再旋Matrix4x4matrix=newMatrix4x4(newVector4(1,2,3,4),newVector4(5,6,7,8),newVector4(9,10,11,12),newVector4(13,14,15,16));Matrix4x4me=Matrix4x4.identity;Debug.Log(m0);Matrix4x4m0=Matrix4x4.zero;Debug.Log(m0);Matrix4x4matrix=newMatrix4x4(newVector4(1,2,3,4),newVector4(5,6,7,8),newVector4(9,10,11,12),newVector4(13,14,15,16));Matrix4x4mi=matrix.transpose;Debug.Log(matrix);Debug.Log(mi);Quaternionquat=matrix.rotation;Debug.Log(quat);Matrix4x4mt=matrix.inverse;Debug.Log(mt);PublicmethodsGetColumnGetacolumnofthematrix.GetRowReturnsarowofthematrix.MultiplyPointTransformsapositionbythismatrix(generic).MultiplyPoint3x4Transformsapositionbythismatrix(fast).MultiplyVectorTransformsadirectionbythismatrix.SetColumnSetsacolumnofthematrix.SetRowSetsarowofthematrix.SetTRSSetsthismatrixtoatranslation,rotationandscalingmatrix.ToStringReturnsanicelyformattedstringforthismatrix.TransformPlaneReturnsaplanethatistransformedinspace.ValidTRSChecksifthismatrixisavalidtransformmatrix.StaticMethodsFrustumThisfunctionreturnsaprojectionmatrixwithviewing.LookAtGivenasourcepoint,atargetpoint,andanupvector.OrthoCreatesanorthogonalprojectionmatrix.PerspectiveCreatesaperspectiveprojectionmatrix.RotateCreatesarotationmatrix.ScaleCreatesascalingmatrix.TranslateCreatesatranslationmatrix.TRSCreatesatranslation,rotationandscalingmatrix.StaticmethodsinMatrix4x4//旋转Matrix4x4mr=Matrix4x4.Rotate(newQuaternion(0.1f,0.2f,0.3f,1.0f));//缩放Matrix4x4ms=Matrix4x4.Scale(newVector3(0.1f,0.1f,0.1f));//平移Matrix4x4mt=Matrix4x4.Translate(newVector3(10f,0f,0f));Matrix4x4matrix=Matrix4x4.TRS(newVector3(1,0,0),Quaternion.identity,newVector3(1,1,1));Debug.Log(matrix);AttitudeandheadingNormalsituation,GimbalLockQuaternionsAquaternionisanalternativemathematicalentitythat3Dgraphicsprogrammersusetorepresentrotations.RotationswithQuaternionstheunitquaternionqcorrespondingtoarotationthroughtheangleθabouttheaxisAisgivenbyconvertaquaternionintotheequivalent3×3rotationmatrix:PropertieseulerAngles:Returnsorsetstheeulerangle.normalized:Returnsthisquaternionwithamagnitudeof1.this[int]:Accessthex,y,z,wcomponentsusing[0],[1],[2],[3].w: WcomponentoftheQuaternion.x: XcomponentoftheQuaternion.y: YcomponentoftheQuaternion.z: ZcomponentoftheQuaternion.publicVector3eulerAngles{get;set;}quaternion=newQuaternion(0.0f,0.0f,0.0f,1.0f);cube.transform.rotation=quaternion;Vector3angles=quaternion.eulerAngles;Quaternion的实例方法Set：Setx,y,zandwcomponentsofanexistingQuaternion.SetFromToRotation：CreatesarotationwhichrotatesfromfromDirectiontotoDirection.SetLookRotation：Createsarotationwiththespecifiedforwardandupwardsdirections.ToAngleAxis：Convertsarotationtoangle-axisrepresentation(anglesindegrees).ToString：ReturnsanicelyformattedstringoftheQuaternion.Quaternionq=Quaternion.identity;q.SetFromToRotation(v1,v2);transform.rotation=q;publicvoidSetLookRotation(Vector3view,Vector3up=Vector3.up);Quaternionquat=newQuaternion();quat.SetLookRotation(Vector3.down,Vector3.left);transform.rotation=quat;Angle:Returnstheangleindegreesbetweentworotationsaandb.AngleAxis:Createsarotationwhichrotatesangledegreesaroundaxis.Dot: Thedotproductbetweentworotations.Euler:Returnsarotationthatrotateszdegreesaroundthezaxis,xdegreesaroundthexaxis,andydegreesaroundtheyaxi