Chapter 13 Vertex and Coordinate System

Transcription

1 [100] 013 revised on cemmath The Simple is the Best Chapter 13 Vertex and Coordinate System 13-1 Class vertex 13-2 Operations for Vertices 13-3 Member Functions for Vertices 13-4 Class csys 13-5 Application Examples 13-6 Summary The class vertex is very useful in handling geometry in the threedimensional space. For example, mesh generation can be carried out by using vertex arrays. This leads to an introduction of another important data type csys which represents a coordinate system. In this chapter, we discuss data types of vertex and csys. Section 13-1 Class vertex Data Structure of Vertex. The vertex in two-dimension or in threedimension is defined as where are the unit vectors in the Cartesian coordinates, and is assumed in two-dimensional space. In Cemmath, the vertex is denoted by v = < x, y > v = < x, y, z > where are double data. In Cemmath, all the vertex data are stored as the 1

2 three-dimensional data. The default format for displaying vertices is < %12.4g %12.4g %12.4g > which can be changed by vertex.format( string ) This can be confirmed by the following Cemmath commands %> default display #> x = 1;; y = pi;; z = sqrt(2);; #> v = <x,y,z>; #> vertex.format(" %15.6e, %15.6e, %15.6e"); v; #> vertex.format("< %12.4g %12.4g %12.4g >"); v; v = < > v = e+000, e+000, e+000 v = < > Writing Coordinate System. The principal coordinate systems are the Cartesian (rectangular), cylindrical and spherical coordinate systems. The polar coordinate system is considered to be the cylindrical coordinate system with. The global Cartesian coordinate is the default coordinate and is designated <x,y,z, rec> which is the same as <x,y,z> // the default writing system is always Cartesian The polar coordinate or the cylindrical coordinate defined as can be applied to write a vertex < r,t, cyl > < r,t,z, cyl > // this is the standard form 2

3 Also, the spherical coordinate defined as can be utilized to write a vertex < R,P,T, sph > // this is the standard Reading Coordinate System. Any point in the global Cartesian coordinate can be read from the cylindrical and spherical coordinate systems by < x,y,z >.cyl < x,y,z >.sph to get and values. For example, #> < 1,1,1 >.cyl ; #> < 1,1,1 >.sph ; result in ans = < > ans = < > At present, it is noteworthy that the angle is written in radian instead of degree. Of course, such a regulation can be easily changed as can be seen later. Conversion between Coordinate Systems. Any vertex written in one coordinate system can be converted to a vertex in another coordinate system by < double,double,double, w_csys >.r_csys < vertex, w_csys >.r_csys < double,double,double, wa_csys[i] >.ra_csys[j] < vertex, wa_csys[i] >.ra_csys[j] where w_csys represents a writing coordinate system, and r_csys a reading coordinate system. In addition, wa_csys represents an array of csys. Then, 3

4 one can easily note that whenever the writing and reading coordinate systems are identical, e.g. < double,double,double, w_csys >. w_csys < double,double,double, wa_csys[i] >. ra_csys[j] are always the same as < double,double,double > The notation in Cemmath is very concise and thus will help users to exploit various coordinate systems with ease. Coordiate Values. Any vertex can be interpreted in one of the Cartesian, cylindrical/polar and spherical coordinates. In Cemmath, the specific coordinate values are designated v.x, v.y, v.z v.r, v.t, v.z v.r, v.p, v.t // ( x, y, z ) for the Cartesian coordinate // ( r, t, z ) for the cylindrical coordinate // ( R, P, T ) for the spherical coordinate It should be noted that the angle system is by default csys.rad based on the radian system in both reading and writing modes. The angle system can be changed by employing degree coordinate system, i.e. csys.deg as can be seen later. An example is as follows. %> coordinate values #> v = < 1,1,1 > ; #> v.x; v.y; v.z; // Cartesian #> v.r; v.t; v.z; // cylindrical #> v.r; v.p; v.t; // spherical v = < > ans = 1 ans = 1 ans = 1 ans = ans = ans = 1 ans =

5 ans = ans = It is also possible to change the specific coordinate value. For example, %> coordinate values #> v = < 1,1,1 > ; #> v.r = 3 ; v = < > v = < > Note that changing the radius in the spherical coordinate does not change the azimuthal and the cone angles. This is true for all other coordinate values, i.e. only one spatial coordinate can be changed by component assignment in each coordinate system. Compound Operations. Compound operations for the vertex elements are very useful to manipulate the positions at the three-dimensional space. For example, a vertex at can be moved to just by v.t += dt. Available compound operations are v.x += d v.x -= d v.x *= d v.x /= d v.x ^= d v.y += d v.y -= d v.y *= d v.y /= d v.y ^= d v.z += d v.z -= d v.z *= d v.z /= d v.z ^= d v.r += d v.r -= d v.r *= d v.r /= d v.r ^= d v.t += d v.t -= d v.t *= d v.t /= d v.t ^= d v.r += d v.r -= d v.r *= d v.r /= d v.r ^= d v.p += d v.p -= d v.p *= d v.p /= d v.p ^= d v.t += d v.t -= d v.t *= d v.t /= d v.t ^= d Examples are %> compound operations #> csys.rad; #> vx = < 2,0.5,1, sph >;; vx.sph ; #> v = vx;; v.r += 1;; v.sph ; #> v = vx;; v.p += 1;; v.sph ; #> v = vx;; v.t += 1;; v.sph ; 5

6 vx = < > ans = < > ans = < > ans = < > ans = < > Section 13-2 Operations for Vertices Binary Operations. Binary operations between vertices are discussed by using two vertices a = < a1, a2, a3 > b = < b1, b2, b3 > Then, binary operations between vertices are defined a + b = < a1+b1, a2+b2, a3+b3 > = a.+ b a b = < a1-b1, a2-b2, a3-b3 > = a.- b a * b = a1*b1 + a2*b2 + a3*b3 a ^ b = < a2*b3-a3*b2, a3*b1-a1*b3, a1*b2-a2*b1 > a.* b = < a1*b1, a2*b2, a3*b3 > a.^ b = < a1^b1, a2^b2, a3^b3 > a./ b = < a1/b1, a2/b2, a3/b3 > a. \ b = < b1/a1, b2/a2, b3/a3 > // element-by-element Examples are %> binary operations #> <1,2,3> + <3,4,5> ; ans = < > #> <1,2,3> * <3,4,5> ; ans = 26 #> <1,2,3> ^ <3,4,5> ; ans = < > 6

7 Note that multiplication of two vertices (i.e. 3D vectors) produces double data not of vertex. Also, note that the hat ^ operator corresponds to the cross product between vertices. These operations are consistent with the typical vector operations In particular, the relational operators are defined a == b is true if a1-b1 + a2-b2 + a3-b3 <= meps a!= b is true if a1-b1 + a2-b2 + a3-b3 > meps a > b is true if a1 > b1, a2 > b2, a3 > b3 a < b is true if a1 < b1, a2 < b2, a3 < b3 a >= b is true if a1 >= b1, a2 >= b2, a3 >= b3 a <= b is true if a1 <= b1, a2 <= b2, a3 <= b3 where meps is a priori prescribed small number. A few examples are as follows. %> binary operations #> <1,2,3> == <3,4,5> ; #> <1,2,3>!= <3,4,5> ; #> <1,2,3> > <3,4,5> ; #> <1,2,3> < <3,4,5> ; #> <1,2,3> >= <3,4,5> ; #> <1,2,3> <= <3,4,5> ; ans = 0 ans = 1 ans = 0 ans = 1 ans = 0 ans = 1 7

8 Binary Operations with double Data. Binary operations between vertices and double data are performed by upgrading double to vertex, i.e. a double data is considered to be a vertex with all the identical components. In summary, bindary operations for vertices are listed as follows v1 == v2 v1 == s s == v2 v1!= v2 v1 > v2 v1 < v2 v1 >= v2 v1 <= v2 v1!= s v1 > s v1 < s v1 >= s v1 <= s s!= v2 s > v2 s < v2 s >= v2 s <= v2 v1 + v2 v1 + s s + v2 v1 - v2 v1 - s s - v2 v1 * v2 v1 * s s * v2 v1 ^ v2 v1 / s s \ v2 v1.+ v2 v1.+ s s.+ v2 v1.- v2 v1.- s s.- v2 v1.* v2 v1.* s s.* v2 v1.^ v2 v1.^ s s.^ v2 v1./ v2 v1./ s v1.\ v2 s.\ v2 Compound Operations. Compound operations for vertices are as follows. v += v2 v -= v2 v ^= v2 v += s v -= s v *= s v /= s Tensor Operations. Vectors in physics, e.g. force, can be also described by vertices in 3D space. In Cemmath, the second-order tensor in physics is treated as a matrix. Then, two vertices, or equivalently two vectors a = < a1, a2, a3 > b = < b1, b2, b3 > 8

9 can be used to define tensor product and tensor division a ** b = [ a1*b1, a2*b1, a3*b1 ] [ a1*b2, a2*b2, a3*b2 ] [ a1*b3, a2*b3, a3*b3 ] a %% b = [ a1/b1, a2/b1, a3/b1 ] [ a1/b2, a2/b2, a3/b2 ] [ a1/b3, a2/b3, a3/b3 ] These operators can be utilized to calculate, for example since we can write the above as df %% da. In addition, the typical multiplication in continuum mechanics such as vector*tensor and tensor*vector can be implemented in Cemmath vertex * matrix matrix * vertex which represent the following operations 9

10 An example for tensor operations is given below. %> tensor operations #> I = < 1,0,0 >;; J = < 0,1,0 >;; K = < 0,0,1 >;; #> va = <1,2,3>; vb = <3,4,5>; va = < > vb = < > #> C = va ** vb ; C = [ ] [ ] [ ] #> D = va %% vb ; D = [ ] [ ] [ ] #> I * D; J * D; K * D; ans = < > ans = < > ans = < > #> D * I; D * J; D * K; ans = < > ans = < > ans = < > Section 13-3 Member Functions for Vertices Member Functions. Avaliable member functions.xrot(t).yrot(t).zrot(t).xrotd(t) rotation around the x-axis rotation around the y-axis rotation around the z-axis rotation around the x-axis in degree 10

11 .yrotd(t) rotation around the y-axis in degree.zrotd(t) rotation around the z-axis in degree.unit normalize by the magnitude.trun truncate.plot plot in 3D space.abs/norm the same as.r.symm(vp) symmetric position with respect to vertex vp are employed to treat vertices in a variety of ways. Section 13-4 Class csys In this section, we discuss the class csys to describe local coordinate system in contrast to the global coordinate systems. Principal Coordinate Systems. The global principal coordinate systems (i.e. the rectangular, cylindrical and spherical) should be treated as special csys. This allows that %> (M #Example ) principal spherical system #> csys.sph; to represent ans = 'spherical' local coordinate system origin = < > dir cosine = [ ] [ ] [ ] Local Coordinate Systems. A local coordinate system is composed of two elements with respect to the global Cartesian coordinate. The first is the origin, and the second is the direction cosines. In the global Cartesian coordinate, the direction cosine of the -axis of a local coordinate can be written as. Similary, the and axes can be denoted by and, respectively. Also, the 11

12 orthogonality requires that whenever is the Kronecker delta. Then, the mathematical definition of a local orthogonal coordinate system is where the origin of a local coordinate system is represented by a vertex, and the direction cosine of a local coordinate system is represented by a matrix of dimension. Then, the coordinate in the local coordinate can be identified as the coordinate in the global Cartesian coordinate. The syntax to create a local coordinate system with an origin at vertex vorg is csys.rec ( vorg, vn1,vn2 ) csys.cyl ( vorg, vn1,vn2 ) csys.sph ( vorg, vn1,vn2 ) or equivalently csys ( 1, vorg, vn1,vn2 ) csys ( 2, vorg, vn1,vn2 ) csys ( 3, vorg, vn1,vn2 ) where the integers 1,2 and 3 denote the rectangular, cylindrical and spherical coordinate, respectively. In the above, two vertices vn1,vn2 are used to generate direction cosines An Example of Local Coordinate Systems. Let us consider a simple example of a local coordinate system by the following Cemmath command %> local coordinate system #> cs1 = csys.cyl( <3,1>, <1,1>, <-1,2>); cs1 = 'cylindrical' local coordinate system 12

13 origin = < > dir cosine = [ ] [ ] [ ] This local coordinate system is shown in Figure 1. Figure 1 A local coordinate system Then, it is easy to find that the point P(1,1,0) in the global coordinate is interpreted to be in the local cylindrical coordinate system. Also the point Q(2,2,0) in the global coordinate is interpreted to be the local cylindrical coordinate system. This can be confirmed by in #> <1,1,0>.cs1 ; // <2, pi*3/4, 0> #> <2,2,0>.cs1 ; // <1.414, pi/2, 0> #> <sqrt(2),pi/2,0, cs1> ; // <2, 2, 0> ans = < > ans = < > ans = < > In the above, writing and reading with respect to a local coordinate system are treated very concisely. Class Functions of csys. Several class functions are 13

14 csys.rec // csys.rectangular, csys.cart, csys.cartesian csys.cyl // csys.cylindrical, csys.polar csys.sph // csys.spherical csys.deg // angles in degree csys.rad // angles in radian (default) csys.pass180/passpi // pass 180 degree, i.e. 0 <= theta < 360 csys.pass0 // pass 0 degree, i.e < theta <= 180 When csys.deg is executed, the degree is adopted to be the angle system. This can be confirmed by %> angle system #> csys.deg; < 1,45,0, cyl >; #> csys.rad; < 1,pi/4,0, cyl >; // angle in degree. csys.deg is activated ans = < > // angle in radian. csys.rad is activated ans = < > The range of angle can be defined in two different ways The first case passes line and the second case passes line. In this regard, two commands csys.pass180/passpi and csys.pass0 can be used to select the range of angle. The default system is csys.pass0, i.e. is assumed. This argument can be confirmed by %> angle range #> csys.pass180; < 1,-1,0 >.cyl; #> csys.pass0; < 1,-1,0 >.cyl; // angle range, 0 <= radian < 2*pi is activated ans = < > // angle range, -pi < radian <= pi, is activated ans = < > Member Functions of csys. Several member functions are.plot // plot coordinate system 14

15 .org.cos // origin of a local coordinate system // direction cosine of a a local coordinate system The origin of a local coordinate system can be changed via member function org. An example is as follows. %> change origin #> cs2 = csys.cyl( <3,1>, <1,1>, <-1,2>); #> cs2.org = < 5,4,7 >; #> cs2; cs2 = 'cylindrical' local coordinate system origin = < > dir cosine = [ ] [ ] [ ] ans = < > cs2 = 'cylindrical' local coordinate system origin = < > dir cosine = [ ] [ ] [ ] However, direction cosine cannot be modified since the orthogonality of the direction cosine need to be preserved. In later version of Cemmath, we will upgrade this part. Referring can be done by %> change origin #> cs2 = csys.cyl( <3,1>, <1,1>, <-1,2>); #> cs2.cos; ans = [ ] [ ] [ ] Section 13-5 Application Examples 15

16 Center of a Triangle. A circle enclosing a triangle is found by vertex operations as follows. %> A circle enclosing a triangle #> va=<3,5,7>; vb=<-1,2,-5>; vc=<4,-6,1>; #> Radius = 0; #> vg=(va+vb+vc)/3; // geometric center va = vb-va; vb = vc-va; vn = va^vb; vx = vn^va; dt = 0.5*((vb-va)*vb)/(vx*vb); vr = 0.5*va+dt*vx; // vr = va/2+t*vx, vr-(vb/2) should be orthogoanal to vb if(radius > 1.e-10) { dt = sqrt( (Radius*Radius-vr*vr)/(vn*vn) ); vr = vr + dt*vn; } #> vcen = va+vr ; #> (vcen-va).abs; #> (vcen-vb).abs; #> (vcen-vc).abs ; // vcen = < > ans = ans = ans = At the end of commands, it is confirmed that the point vcen is indeed the center of a triangle. Array of Vertex. A number of vertices can be created by an array of vertex, and they can be used to generate meshes. An example is to make an annular mesh as follows. %> Vertex array to generate annular mesh 16

17 #> vertex V[50]; #> csys.deg; #> for.i(0,4) { r = *i;; for.j(0,9) { t = 36*j;; V[10*i+j] = < r,t, cyl >;; } } #> V; V = vertex [50] [0] = < > [1] = < > [2] = < > [3] = < > [18] = < > [19] = < > and more... (50 elements) Section 13-6 Summary Class Functions of vertex. vertex.format( string ) Conversion between Coordinate Systems. Any vertex written in one coordinate system can be converted to a vertex in another coordinate system by < double,double,double, w_csys >.r_csys < vertex, w_csys >.r_csys Member Functions of vertex..xrot(t).yrot(t).zrot(t).xrotd(t) rotation around the x-axis rotation around the y-axis rotation around the z-axis rotation around the x-axis in degree 17

18 .yrotd(t) rotation around the y-axis in degree.zrotd(t) rotation around the z-axis in degree.unit normalized by the magnitude.trun truncate.plot/plot3 plot in 3D space.abs/norm the same as.r.symm(vp) symmetric position with respect to vertex vp Class Functions of csys. csys.rec // csys.rectangular, csys.cart, csys.cartesian csys.cyl // csys.cylindrical, csys.polar csys.sph // csys.spherical csys.deg // angles in degree csys.rad // angles in radian (default) csys.pass180/passpi // pass 180 degree, i.e. 0 <= theta < 360 csys.pass0 // pass 0 degree, i.e < theta <=180 csys.rec ( vorg, vn1,vn2 ) csys.cyl ( vorg, vn1,vn2 ) csys.sph ( vorg, vn1,vn2 ) Member Functions of csys..plot/plot3.org.cos // plot coordinate system // origin of a local coordinate system // direction cosine of a local coordinate system // // end of file //