A Derivation of Free-rotation in the Three-dimensional Space

Transcription

1 International Conference on Adanced Information and Communication echnology for Education (ICAICE 013) A Deriation of Free-rotation in the hree-dimensional Space Legend Chen 1 1 Uniersity of Shanghai for Science and echnology optoelectronic information and Computer Engineering Shanghai China Abstract he paper describes an algorithm deriation of rotation in the three-dimensional space, which is an efficient method for rotating objects in the space. he key in the deriation process is to summarie the many thoughts on this issue, included the geometric algebra, coordinate transformation, linear space and matrix operations, and analysis of some possible algorithm of rotation, such as basic rotation, the combination of rotations and coordinates system transformation. Finally, obtains a matrix as a formula for the free-rotation in the three dimensions. Keywords: hree-dimensional, Freerotation, Computer Graphics, Coordinate ransformation he rotation in two-dimensions is easy to draw, but also well understood. Howeer, How it be extended to the freerotation in the three-dimensions? 1. Basic Rotation in a Plane Using triangle transformation formula, you can get the following formula for a rotating basis. X cos Y sin X X sin Y cos Y he formula is easy to conclude from the triangle transformation by the following equation. rcos( ) X rsin( ) Y And, r cos X r sin Y r cos cos r sin sin X r cos sin r sin cos Y θ is the angle of rotation, and α is the angle of ector. his formula is a ery classical method of rotation in the two-dimensional space. Parameters are the angle and ector, return a coordinate ector, ery concise.. Continue to Rotate in the Space In three dimensions space, using the known formula preiously mentioned, corresponding to calculation on the YZ plane, XZ plane and XY plane, the rotation around the X axis, Y axis and Z axis can be resoled directly. hus, there is a complex method to complete a free rotation with a series of free rotation, which seems feasible. After all, we just want to find a plane can be rotated. Based on the main idea of the preious section, the algorithm is as follows: Defines a ector P, that the location and direction of rotation axis he authors - Published by Atlantis Press 49

2 Step 1: ranslate P, so that the starting point of P coincides with the coordinate origin Step : Rotate P around the X axis, which another point of P is to the XZ plane Step 3: Rotate P around the Y axis, which another point of P coincides with the Z axis Step 4: Rotate P around the Z axis, which Step 5: Y axis reerse, inerse transformation Step 6: X axis reerse Step 7: ranslation reerse his is definitely a product of a lot of triangle rotations and translations. It is undeniable that there are too many ariables to calculate the intermediate. What s worse, no efficiencies, the accuracy is also unknown (be mentioned later and gie a proof). 3. Gain a New Coordinate System Following the aboe ideas, the new problem lies in the coordinate system transformation. We need to create a new coordinate system based on one shaft, and get new space coordinate information. Imitation of pinhole imaging, just a way, can be used to simulate the new coordinate system. Of course, you could also simulate a projector, but soon you will find that it is merely a geometric algebra approach and a quite timeconsuming processing. Based on the geometric algebra, the approach is as follows: Step 1: Using operation of dot product and cross product with right-hand rule, obtain three orthogonal ectors presenting new coordinates. 1 1g is random ector, not parallel to 1. Conert them to unit ectors. hen, the coordinates consist of unit ectors of 1, and 3. Step : Comparison of the point in the space with the coordinates, calculate their relatie displacement. Defines ectors of 1,, 3 and 0 in the new coordinates, that 1 as a new X axis, as a new Y axis, 3 as a new Z axis and 0 as a relatie position. 0g3 0 0 ( 0 ) cos sin 0 3 0g g 0 1,, 3 alue is 1. he r is the distance between the point with the new origin. Results x y rcos rsin 3g0 3 It is easy to understand, but the inerse transformation is impossible in the rotation transformation. Its calculation still spends so much, which not will be used in the practical application. 50

3 4. ransformation of Coordinate System Now we come to the point. In order to express clearly, I use linear algebra through the matrix to sole the problem of coordinate transformation, Simply, skip shift thing. In other word, the new coordinate system origin is the origin of the old coordinate system. he following matrix presents a normal coordinate system As long as it is an orthogonal matrix, each column ector can be regarded as a sub-ector of a coordinate axis. hey are linearly independent, orthogonal a a b a 0 b 1 c 0 b 0 0 1c c For a space ector, a is the alue of X axis component, b is the alue of Y axis component, c is the alue of Z axis component, and the combination of these components is the ector itself. Suppose there is a point in a new coordinate system corresponds to the other point in the space coordinate system. a11 a1 a13 x a a a a y b a a a c Because it is orthogonal matrix, the following formula established hen 1 AA AA E x a a a a a a a a y a a a b a a a b a31 a3 a c a31 a3 a c Considering the case of translation, the formula can be expressed in a different sense, as the following two. x a11 a1 a13 a u y a a a ( b ) 1 3 a31 a3 a c w a a11 a1 a13 x u b a a a y 1 3 c a31 a3 a w Note: (u,, w) as a translation ector 5. Rotation in a New Coordinate System (1) () According to the rotation axis coincides with the new Z axis, we create a new coordinate system (presented as a matrix A if it s exited, which certainly exited), rotate it, and then conerted back to the original coordinates. Here is an example. Equation of plane rotation x cos sin 0 x y sin cos 0 y here is a -axis rotation matrix, and θ is the angle of rotation. o apply the two formulas deried preiously (1), () x x u x a u y Ay, y A ( b ) w c w 51

4 ( a, b, c ) as the initial point. ( x, y, ) as a the point in new coordinate system. ( x, y, ) as a point after a rotation. ( x, y, ) as a result point. x cos sin 0 a u u y A sin cos 0 A ( b ) c w w he preiously mentioned method with a series of axis rotating, it is also obious correctness, which are all orthogonal transformations. x a u u 1 1 y X Y Z Z Y X 1 ( b ) c w w 6. Optimiation for the Algorithm Coordinate transformation method can be used to get answers to the aboe issue of this free rotation. Howeer, in the aboe mentioned transformation matrix, the first column and second column ectors are no other special requirements, just compliance with the orthogonal condition and unit condition. Of course, the frame of reference really has nothing to do with the rotation. By calculating, we should be able to replace the intermediate ariables to achiee the purpose of simplifying the formula equation. As a coordinate system transformation matrix a1 b1 c1 A a b c a3 b3 c3 hen, the rotate equation r11 r1 r13 cos sin 0 r1 r r3 A sin cos 0 A r31 r3 r In accordance with the conditions of the orthogonal relationship, their dot product and cross product is ero. a1b1 a b a3b3 0 c1b1 c b c3b3 0 a1c1 a c a3c3 0 a1a1 a a a3a3 1 b1b1 b b b3b3 1 c1c1 c c c3c3 1 Addition of the right-hand rule r11 : a b3 a3b c1 a3 b1 a1 b3 c a1b a b1 c3 c a1 a1 s a1 b1 s a1 b1 c b1 b1 c1 c1 1c c1c1c r1 : c a1a s a1b s a b1c b1b c1c s c3 1c c1c r31: c a1a3 s a1b3 s a3b1c b1b3 c1c3 s c 1c c1c3 herefore, the final result of the transformation matrix can be calculated as (3). Its parameter is only a c ector, the direction ector of the rotation axis. 5

5 sin c (1 cos ) c1 c3 sin c1 (1 cos ) c c3 (1 cos ) c3 cos (1 cos ) c1 cos sin c3 (1 cos ) c1 c sin c (1 cos ) c1 c3 sin c3 (1 cos ) c1 c (1 cos ) c cos sin c1 (1 cos ) c c3 (3) 7. Conclusions he final formula for the free-rotation in the three dimensions is Simple and easy to use. he thinking of the deriation plays a significant role in the whole transformation process. Reference frame transformation has been applied to sole many complex problems such as transform, w transform. 8. References [1] G. E. Shilo, Linear Algebra, Prentice Hall, [] D. F. Rogers, Procedural Elements for Computer Graphics, China Machine Press, 00. [3] J. W. Brown, and R. V. Churchill, Complex Variables and Application, China Machine Press, 004. [4] M. Gross, and H. Pfister, Pointbased graphics, Elseier,