Rotation about an arbitrary axis and re ection through an ...

1 Rotation about an arbitrary axis andreflection through an arbitrary planeEm d Kov csDepartment of Information TechnologyEszterh zy K roly April 22, 2012 Accepted November 7, 2012 AbstractThe aim of this paper is to give a new deduction of Rodrigues rotationformula. An other benefit of the this deduction is to give a transforma-tion matrix of reflection through an arbitrary plane with the same deductionmethod. In our opinion this deduction method is better for students, whoare learning computer :Point transformation, Transformation Matrix, Rotation , Reflec-tion, Rodrigues Rotation formula,MSC:Primary 68U05, Secondary 65D181. IntroductionIn the theory of three-dimensional (3D) Rotation Rodrigues Rotation formula (see[7]) is an efficient matrix for rotating an object around arbitrary axis .

2 In this paperwe will deduct the matrix form in a different way from the well known method whichis published in Rodrigues paper [7], cited in Johan s paper [6] and also described inWolfram Mathworld site (see [1]). First we give a short introduction of linear pointtransformation, then we inroduce a new deduction of reflection about an arbitraryaxis. Next, we will prove, that our matrix is analogous to the original Rodrigues formula. In section three, we describe a matrix of reflection through an arbitraryplane, which is a consequence of our Mathematicae et Informaticae40(2012) pp. 175 186 Linear Point TransformationThree dimensional point transformation is one of the well known computer graphicsmethods, when we manipulate the points of objects, like rotate, translate and on the advantages of homogeneous coordinates, 3D transformations can berepresented by4 4matrices (see [2] and [3]).

3 Generally the following matrixequation describes the point =M p,( ) x 1x 2x 3x 4 = m11m12m13m14m21m22m23m24m31m32m33m34m41m 42m43m44 x1x2x3x4 .When we use more 3D transformations after each other, it is constructed of matrixmultiplications (see [8]), therefore composition of 3D transformation can be repre-sented by the multiplication of transformation matrices. The order of multiplicationdepends on the original form of matrix equation, since the matrix multiplicationwas noncommutative operation. If we multiply the point from left with the trans-formation matrix in Eq. ( ), then we must multiply the transformation matricesin reverse order. LetsM1the first,M2the second transformation matrix, thenp =M1 p,p =M2 p.

4 Using the associative property it becomesp =M2 (M1 p) = (M2 M1) we multiply the matrices in reverse orderM3=M2 M1,from thatp =M3 the point is multiplied by the transformation matrix from the right, then it meansthe equivalent system. Some graphics library, DirectX use the latter method,in this case these system use the transposed this paper we deal with the general case of Rotation about an arbitrary axisin space. It frequently occurs e. g. in robotics, animation and Rotation about an arbitrary axisIf we want to construct Rotation about an arbitrary axis , then we have a good solu-tion namely Rodrigues Rotation formula, see [1] on Wolfram MathWorld site. Lotsof literatures and internet sources give this method.

5 The problem is that the math-ematical deduction is not suited for the previous section from methodical of students could not understand the mathematical deduction of Rodrigues formula, which is presented on the Wolfram MathWorld site, and therefore some176E. Kov csof them could not use it. When we teach basic point transformations and we tryto extend it towards the composition of 3D transformations, then it could be agood example about the Rotation about an arbitrary axis . It would be better if wecan give the Rodrigues Rotation matrix with the composition of basic linear pointtransformations, and apply multiplication of transformation matrices. In this pa-per we deduce the Rotation matrix and prove the computed matrix is an equivalentof the Rodrigues formula.

6 Anyone can find the deduction in Rogers s textbook [8],but now we continue the basic idea is to make the arbitrary Rotation axis coincide with one of thecoordinate axis . Assume an arbitrary axis in space passing through the pointP0(x0,y0,z0)andP1(x1,y1,z1).Figure 1: Rotation about an arbitrary axisIn this case Rotation about this axis by some angle is accomplished using thefollowing procedure:1. Translate theP0(x0,y0,z0) axis point to the origin of the coordinate Perform appropriate rotations to make the axis of Rotation coincident withz-coordinate Rotate about thez- axis by the angle .4. Perform the inverse of the combined Rotation Perform the inverse of the about an arbitrary axis and reflection through an arbitrary plane177 For the simplicity we compute theu=P1 P0vector, which after the normal-ization can give us the direction cosines of axis :ue:=u|u|= (cx,cy,cz).

7 In Fig. 2 the direction cosines are satisfied the following equation:c2x+c2y+c2z= 1,cos x=cx,cos y=cy,cos z= 2:Direction cosinesThe required translation matrix isT( p0) = 1 0 0 x00 1 0 y00 0 1 z00 0 01 .In the next step the procedure requires two successive Rotation about thex-axisby the angle xandy- axis by the angle the Rotation around the thex-axisthe original Rotation axis will be in the[x,z]coordinate pane. (See Fig. 3).From the Fig. 3 comesd= c2y+c2z, and we do not calculate explicitly theangle x, because we only use its sin and cosine values in the Rotation matrix:sin x=cyd,cos x= Kov csFigure 3: Rotation aroundx-axisThe Rotation matrix isRx( x) = 100 00cz/d cy/d00cy/d cz/d0000 1 .( )We can get the second Rotation matrix in a similar way, where we rotate aroundthey- axis by angle 4: Rotation aroundy-axisRotation about an arbitrary axis and reflection through an arbitrary plane179 From the Fig.

8 4 comessin y=d,cos y= Rotation matrix is with negative directionRy( y) = d0 cx00 10 0cx0d00 00 1 .( )The complete transformation isM=T 1( p0)R 1x( x)R 1y( y)Rz( )Ry( y)Rx( x)T( p0),( )where the upper index 1means the inverse transformation, soM=T(p0)Rx( x)Ry( y)Rz( )Ry( y)Rx( x)T( p0),( )where we used the reverse multiplication order as we mentioned in the previoussection. The computation is finished at this point in Rogers s textbook [8].Now we are giving one of our new results, and in the section we are provingthe formulas with Maple computer algebra formula in ( ) can be enough, if someone only use the basic transformationmatrices in the matrix class of the graphics engine. But methodically for thebetter understandability and based on our students searching practice in internetliterature, we must continue the multiply the inside five matricesR=Rx( x)Ry( y)Rz( )Ry( y)Rx( x),( )andM=T(p0)RT( p0).

9 Consider that the inverse of Rotation matrix equals with the transposed matrix,we getR=RTx( x)RTy( y)Rz( )Ry( y)Rx( x).( )In the next section we are going to prove that if we expand the matrix multipli-cation in Eq. ( ), then we get the general Rodrigues form. In [9] or in [1] we canfind the totally different deduction of the Rodrigues form, but as we mentionedwe are not satisfied the authors deduction way, therefor we give a new Maple CAS is very robust and efficient tool for calculating multiplication oftransformation Kov Proof with MapleThe main problem of the proof is that multiplication of five matrices in Eq. ( ).In order to correct calculation we used Maple computer algebra system (CAS). In[4] and [5] the author explains why Maple is a useful tool for teaching computergraphics in higher education.

10 In Eszterh zy K roly College we use CAS software inteaching undergraduate students studiing Software Information Technology bach-elor can use the power of the linalg package of Maple, to easily multiply the Maple command is> rod:=simplify(Transpose(RX).Transpose(RY ). );where we used that the transposed Rotation matrix equals the inverse of therotation matrix, and the rod means the Rodrigues form. After we used thebuilt-insimplifyfunction we got the output in Fig. 5:First result in MapleThe computed formula is extremely complicated. So we must look for otherRotation about an arbitrary axis and reflection through an arbitrary plane181simplification possibilities. We can use the combination ofsimplifyandsubstitutionfunctions repeatedly:f o ri from 2 to 3dof o rj from 2 to 3dorod [ i , j ]:= s i m p l i f y ( subs ({ cx^2=1 (cy^2+cz ^2)} , rod [ i , j ] ) ) ;od ;od ;We can see the output in Fig.