Math Is Fun Forum

i can remember in worms armageddon (a game) if you had the swear filter on, then the f-word would be changed to 'love' when i reinstalled i forgot to disable it, and i was becoming quite confused

whenevr anyone would say f*** you, i would keep seeing love you, and i was getting a bit disturbed (in the context it was being used)

328. In ΔPQR, side QR=10 cm, height PM=4.4 cm and side PR = 11 cm. What is the length of altitude QN?

whats M ?

329. The area of a square with side x is equal to the area of a triangle with base x. What is the altitude of the triangle?

What is the altitude of an equilateral triangle of side 2√3 cm?

done and done, ive rewritten the first post, using the uploaded images.

Actually, this is a very messy program i quickly put together just to test the torus rendering, the source code is 10kb for the main cpp without about another 10kb of my own code for files containing a Matrix class, a vector class, and classes i made for loading bitmaps and for solving quartic equations.

the pink image is the texture im applying to the surface, i just used smudge tool in photoshop and added noise

well, im not so sure about that, via ray tracing, id say deffinately not since it can take depending on the scene seconds or days to render a scene, the only i could see that happening is to have the editing enviroment have a traditional scan line rendered scene aswell that you can edit visually. But thats along way away Although ill deffinately think about it

to the flash application, what its actually doing, is generating the quartic function of a ray-torus intersection, and then just graphing it, finding the roots and displaying them via the blue vertical lines (did you click it to regenerate?)

http://img60.imageshack.us/img60/4500/showsub7da.gif

that is sort of showing what the sub division thingy is doing

it takes the xmin and xmax and finds the y values
it then splits this in two calling the same function with xmin and the mid of xmin and xmax, and then with the mid of xmin and xmax with xmax so that it then calls the function for each half, and then keeps repeating so it then cuts to quarters, then eights, then 16ths, until the number of sub divions has reached the max. Which in the case is 3

so once the number of sub divisions is the desired, in this case 3, it tests the y-coordiantes, and if they are on opposite sides of the x-axis, uses the mid of the two x values as the root, so you can see in the image that in this case, with just 3 sub divisions, the 2 roots on the right are quite accurate, but not the one on the left, if you increased the number of sub divisions, the roots become more and more accurate

infact, i had it so stuck in my head to use a  recursion method, that i can probably just make it run faster by just calculating the x-increment needed, and just looping from xmin to xmax, but nevermind

the good thing about this method, is that i can maybe make some culls to the sub divions, like if the y values are very large and on the same side, depending on the current sub division, stop sub dividing for that sector because its very likely there wont be a root or something

well i found one last error in my quartic coeffecients, and ive tested it out by finding a coordinate on the torus via the parametric equations and given a vector and a calculated value for 't' found the value of the quartic, for a point on the torus, the value of the quartic is < 1 * 10^(-4), and i tried increment 't' by a small amount like 0.001 and the value suddenly jumped up to >1*10^6

so it deffinately seems to be correct now

(my mistake was in the t coeffecient, i had 2(a•v)(2|a|^2+ R^2 - r^2), where it should have been 4(a•v)(|a|^2+ R^2 - r^2) )

however, i did a bit of looking on the internet, and it seems the general solution for a quartic is very numericaly unstable, which i seem to have found out on my own, so im thinking of other ways to go about solving it, the only way ive seen is a recursive algorithm to find the roots by basicly subdividing a search area between randomely selected values, until a value is found that the quartic is less than a given range

but if someone get find a numericaly stable method of solving a quartic, itd be much appreciated

i only usually do when im figuring something out and asking at the same time, and then end up answering myself before anyone responds

null and void

a dot b = cosine of angle between them, cosine of 90 degrees is 0, so if a dot b is 0, they are perpendicular

a dot b = 2*5 + 3*-2 + 1*-4 = 10-6-4 = 0, a,b are perpendicular
a dot c = 2*10 + 3*-13 + 1*19 = 20 - 39 + 19 = 0, a,c are perpendicular
b dot c = 5*10 + -2*-13 + -4*19 = 50 + 26 - 76 = 0, b,c, are perpendicular

well, its the third one im building, i started with a very simple one having only reflections, spheres, ellipsoids and planes
then another smaller one that had full camera and transformations for the primitaves

the new one im making is a MASSIVE advancement on it, along with a full GUI im creating for it to encapsulate all its tools

this one will have spheres,planes,cones,tori,cylinders,capsules (all rotateable and scalable) (all with textures, normal mapping, refraction maps, reflection maps (i.e. you can have a sphere with a procedurally generated fractal texture, with the same texture renderered in grey scale for a reflection map so that the white parts will reflect all light, and the black parts reflect no light, with a heightmap converted to a normal map for normal mapping to give it a 3d look, with a refraction map opposite to the reflectoin map so that the black parts on the reflection map would allow all light to pass through and be refractad))

along with a generic quadric and quartic shape primitave, along with procedurally generated Julia quaternion fractals (along with other types) and procedural 3d Perlin noise clouds/solids  along with 3d procedural textures)

along with CSG so that you can have a sphere with a torus cut out of it to give a round indent on the sphere) along with convex polygons.

along with 3d heightmap terrain, using another type of texture of colour which holds data for each vertex of the percentages of each 8 textures per heightmap texture to be used at that vertex)

and if i learn it, photon mapping.  Also fuzzy reflections (i.e. multiple rays cast out to give appearance of noise at a very low level) which i might use for 'fake' photon mapping to scatter rays of light from a scenario when light would be absorbed to give atleast some sort of real ambient lighting in reverse.

Tori, especcialy from the quartics is very important i feel, since using CSG, you can make mega complex shapes, using spheres, toruses, planes and cylinders.

I have already written in a seperate c++ program, a compiler that takes a scene file, containing a function based language to set up the scene that parses and converts it to a binary .drt file read by the renderer which will save a .bmp of the rendering to the given directory

a pretty BIG project, but its constantly challenging me in ways of maths, programming and algorithmics so its never a bore and since ive started ive pretty much learnt/figured out most of what ive just written above (id already written programs in the past to create fractals and quaternion renderings, but never ray traced or included in a ray tracing package)

this is all alongside my GCSE and A-Level exams (GCSE + A-Level Maths of which im studying 2 years early)

ANYWAYS, in terms of the toruses
rays are cast out from the camera in its system, then transformed and translated to world system, then translated and transformed into the objects base system so that the object is always at the centre, un rotated and un scaled so that the intersections can be found with far less complex math, and then intersection point and normal to surface at the point are un-transformed and un-translated back into world space