Quaternion Interpolation in Forth?

jrh wrote: > I am looking for a Forth version of interpolation of 3D angles. > Any idea where one might be found? > > jrh Look at http://www.taygeta.com/fsl/library/quaternions.f -- Julian V. Noble Professor Emeritus of Physics University of Virginia

"Julian V. Noble" <jvn@virginia.edu> wrote in news:ekv91p$o7o$1 @murdoch.acc.Virginia.EDU: > http://www.taygeta.com/fsl/library/quaternions.f Thanks for the link, The problem is to develop a fast method of determining if a line from the apex of a triangular pyramid, goes through the base. If it does determine the x y z interpolation constants otherwise repeat the process. The quaternion code isn't exactly what I need, but it's a starting point. The goal is to simulate a Physics problem, using statistical multiplexing. John Hart

On Sun, 3 Dec 2006 22:59:39 -0800, jrh wrote (in article <Xns988EF416EEFC1nospamcom@69.28.173.184>): > The problem is to develop a fast method of determining if a > line from the apex of a triangular pyramid, goes through > the base. If it does determine the x y z interpolation constants > otherwise repeat the process. Triangular pyramid. Is that a tetrahedron? I'm trying to picture the problem. Isn't there a simple vector algebra solution? It seems, granted just a guess, that you could do it all with dot products, which is what computers love to do. And xyz interp constants? Something to do with where it pierces the base? Approximate direction cosines as rational numbers perhaps? If the apex is at the origin and the base "down" perhaps a comparison of direction cosines of the three legs and the line. -- Charlie Springer

"jrh" <no@spam.com> a �crit dans le message de news: Xns988EF416EEFC1nospamcom@69.28.173.184... > "Julian V. Noble" <jvn@virginia.edu> wrote in news:ekv91p$o7o$1 > @murdoch.acc.Virginia.EDU: > >> http://www.taygeta.com/fsl/library/quaternions.f > > Thanks for the link, > > The problem is to develop a fast method of determining if a > line from the apex of a triangular pyramid, goes through > the base. If it does determine the x y z interpolation constants > otherwise repeat the process. > Another method is to use the faces planes equations: Use normalised planes equations, such as the planes normals directions are inside the pyramid. Base Fb(x,y,z) = a*x + b*y + c*z + d = 0 For 3 (or more, but convex) faces: F1(x,y,z) = a1*x + ... +d1 = 0 F2(x,y,z) = a2*x + ... +d2 = 0 ... Fn(...) = an ........+dn = 0 Solution: find line intersection with base: point Pi = (Xi, Yi, Zi) FOR i=1 to n | IF Fi(Xi, Yi, Zi)<0 return outside return inside (or limit epsilon ?) The problem is a bit complicated if the polyhedron is not convex, but the method is the same. Charles > The quaternion code isn't exactly what I need, but it's > a starting point. The goal is to simulate a Physics problem, > using statistical multiplexing. > > > John Hart

"Celime" <cm175872@scarlet.be> a �crit dans le message de news: ApWdnR-w-ZnUQe7YRVnyuA@scarlet.biz... > for clarity: Solution: find line intersection with base: point PI = (XI, YI, ZI) FOR j=1 to n | IF Fj(XI, YI, ZI)<0 return outside return inside (or limit epsilon ?) It's funny to notice that a plane equation is like quaternion with another interpretation for the independent term. plane: d = distance to origin. quaternion: d = rotation around (a,b,c) axis.

Celime wrote: ... > The problem is a bit complicated if the polyhedron > is not convex, but the method is the same. Can a tetrahedron be concave? Jerry -- Engineering is the art of making what you want from things you can get. ��

Charlie Springer <RAM@regnirps.com> wrote in news:0001HW.C1990C960038E9DBF04075B0@news.nw.centurytel.net: > On Sun, 3 Dec 2006 22:59:39 -0800, jrh wrote > (in article <Xns988EF416EEFC1nospamcom@69.28.173.184>): >> The problem is to develop a fast method of determining if a >> line from the apex of a triangular pyramid, goes through >> the base. If it does determine the x y z interpolation constants >> otherwise repeat the process. > Triangular pyramid. Is that a tetrahedron? A tetrahedron is a triangular pyramid with all edges equal length. The three base edges are equal length and edges to the apex are approximately twice as long. > I'm trying to picture the problem. Isn't there a simple vector > algebra solution? It seems, granted just a guess, that you could > do it all with dot products, which is what computers love to do. > And xyz interp constants? Something to do with where it pierces the > base? Approximate direction cosines as rational numbers perhaps? The set of constants are going to be applied to the triangle created by a plane passing through a pyramid to determine where the line intersects the plane. > If the apex is at the origin and the base "down" perhaps a comparison > of direction cosines of the three legs and the line. jrh

"Jerry Avins" <jya@ieee.org> a �crit dans le message de news: FvmdneNxrZUQ0OnYnZ2dnUVZ_rOdnZ2d@rcn.net... > Celime wrote: > > ... > >> The problem is a bit complicated if the polyhedron >> is not convex, but the method is the same. > > Can a tetrahedron be concave? No, for a tetrahedron, it is not possible. > > Jerry > -- > Engineering is the art of making what you want from things you can > get. > ��

jrh wrote: > Charlie Springer <RAM@regnirps.com> wrote in > news:0001HW.C1990C960038E9DBF04075B0@news.nw.centurytel.net: > >> On Sun, 3 Dec 2006 22:59:39 -0800, jrh wrote >> (in article <Xns988EF416EEFC1nospamcom@69.28.173.184>): > >>> The problem is to develop a fast method of determining if a >>> line from the apex of a triangular pyramid, goes through >>> the base. If it does determine the x y z interpolation constants >>> otherwise repeat the process. > >> Triangular pyramid. Is that a tetrahedron? > > A tetrahedron is a triangular pyramid with all edges equal length. That's a /regular/ tetrahedron. > The three base edges are equal length and edges to the apex are > approximately twice as long. So it's not regular. It's still bounded by four planes, and that's what makes that particular polyhedron a *tetra*hedron. ... Jerry -- Engineering is the art of making what you want from things you can get. ��

Celime wrote: > "Jerry Avins" <jya@ieee.org> a �crit dans le message de news: ... >> Can a tetrahedron be concave? > > No, for a tetrahedron, it is not possible. Thanks. That's what I thought. Jerry -- Engineering is the art of making what you want from things you can get. ��

jrh wrote: > Charlie Springer <RAM@regnirps.com> wrote in > news:0001HW.C1990C960038E9DBF04075B0@news.nw.centurytel.net: > >> On Sun, 3 Dec 2006 22:59:39 -0800, jrh wrote >> (in article <Xns988EF416EEFC1nospamcom@69.28.173.184>): > >>> The problem is to develop a fast method of determining if a >>> line from the apex of a triangular pyramid, goes through >>> the base. If it does determine the x y z interpolation constants >>> otherwise repeat the process. > How about this: 1. project the line through the x-y (base) plane. 2. locate the point of intersection (i.e. the coordinates at z=0 in the x-y plane). 3. determine whether that point is inside or outside the base triangle. There is a simple algorithm for #3. Easiest in complex arithmetic. So I fail to see the need for quaternions in this application. -- Julian V. Noble Professor Emeritus of Physics University of Virginia

Julian V. Noble wrote: ... > There is a simple algorithm for #3. Easiest in complex arithmetic. > So I fail to see the need for quaternions in this application. Dot product? Jerry -- Engineering is the art of making what you want from things you can get. ��

Jerry Avins wrote: > Julian V. Noble wrote: > > ... > >> There is a simple algorithm for #3. Easiest in complex arithmetic. >> So I fail to see the need for quaternions in this application. > > Dot product? > > Jerry In 2 dimensions, if w = u+iv and z = x+iy represent 2 vectors in the Argand plane, then conjg(w) * z contains both the dot (real part) and cross (imaginary part) products of the vectors. I will put a version of my upcoming CiSE column (March/April 07) on complex arithmetic on my web page so you can see how to find out if a point is inside or outside a given triangle in the plane. -- Julian V. Noble Professor Emeritus of Physics University of Virginia

"Julian V. Noble" <jvn@virginia.edu> wrote in news:el7bij$3ei$1 @murdoch.acc.Virginia.EDU: > Jerry Avins wrote: >> Julian V. Noble wrote: >>> There is a simple algorithm for #3. Easiest in complex arithmetic. >>> So I fail to see the need for quaternions in this application. >> Dot product? >> Jerry > In 2 dimensions, if w = u+iv and z = x+iy represent 2 vectors > in the Argand plane, then conjg(w) * z contains both the dot > (real part) and cross (imaginary part) products of the vectors. > I will put a version of my upcoming CiSE column (March/April 07) > on complex arithmetic on my web page so you can see how to find > out if a point is inside or outside a given triangle in the plane. Quaternions may still be necessary. The Orientation is polar, and positional inference is (xyz). The following description will provide some context. ----------------------------------------------------------------- Table for the statistical multiplexing of viewing cones. The STAT-MUX-TABLE is used by all objects is rebuilt once each cycle. The objects position is added to the values in the table to compute the addresses to be sampled for each object. The sample location is the intersection of a random angle and a random circle in the viewing cone a random distance from the origin The viewing cones axies are defined by the SENS-ANG table. Mathematical constructs: The rectangular to polar transformation of space. ----------------------------------------------------- . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. / . . / . / . z . / . y / / ' ' ' '.' ' ' ' '/' ' ' ' '/ \ ~ ~ ~ ~/~ ~ ~.~ ~ ~ ~ ~ ~ ~\ / \ / . \ / \/_ _ _ _ _ _ _ _ _ _ _ _ _ _\/ o x | . | . | . | . | . | . | . | . | . | . |.________________________ |.__________ x^2 + y^2 + z^2 = p^2 TAN a = x/y ? y/x TAN b = z/c ? c/z x = p sin b cos a y = p sin b sin a z = p cos b 1. status.aob --- a angle of object ( orientation ) 2. status.bob --- b angle of object ( orientation ) 3. consta.asn --- a angle of sensor ( facet ) 4. consta.bsn --- b angle of sensor ( facet ) 5. lookup.pac --- plane of acquisition cone ( orientation + facet ) 6. lookup.aac --- angle of acquisition cone ( orientation + facet ) 7. random.dst --- distance to acquisition circle ( DIST:DSTB) 8. random.rad --- radius of acquisition cone ( RADI:DSTB) 9 calclu.dia --- diameter of acquisition circle ( DIAM:DSTB) 10. random.arc --- arc to viewing point on acquisition circle The procedure to build the statistical multiplexer table. Get pac, get aac ( axis, from SENS-ANG table ) Get random distance, random arc and random radius. Scale random radius. ( random distance ) Calculate the xyz of the point a random distance along the axis. Calculate the xyz of the point a random angle on the viewing circle. Add the xyz's. Angle on plane Rotation of the plane VERTICES: A B 0 0 31.717474411461 0 72 144 216 288 63.434948822922 0 36 72 108 144 180 216 252 288 324 90 0 36 72 108 144 180 216 252 288 324 116.565051177078 0 36 72 108 144 180 216 252 288 324 148.282525588539 0 72 144 216 288 180 0 Define a plane for each facet of the 3x icosahedron Convert sensor direction into line from apex. (ax +by +cz = 0) Find where the direction intersects with the plane. rotation change? if select interpolation set last_sensor first_sensor do begin calculate sensor line intersection point on base plane of first facet. point inside facet? until calculate offset_vector facet i save in table loop then last_sensor first_sensor do i get facet get offset_vector ------------------------ ICOSAHEDREN_FACET_TABLE LINES from origin SAMPLE_DISTRIBUTION_TABLE LINES from origin rebuilt each generation STATISTICAL_MUX_TABLE SCALES sensors focus (60sensors) SAMPLE_INTERPOLATION_TABLE VECTORS rebuilt when object rotates The viewing cone for all 60 sensors is about 60 degrees, each icosahedron facet is divided into 64 triangles. 1280 total facets. interpolation points are the center of each facet. 1. Find the facet. (triangle the sensor vector passes thru) facet_list = 1280(x1.y1.z1,x2.y2.z2,x3.y3.z3) sensor_vector = (x.y.z) 2. Find the intersect point. facet_pointer = (1..1280) 3. Calculate the distance (xyz) between the intersect and the center of the facet. (translation vector) (save in table, update each time orientation changes) 4. Scale the translation vector by the distance the random sample plain is from the origin. 5. Add the translation vector to the random sample point. ----------------------------------------------------------- copyright John R Hart