c++ - How to infer translate, shear, etc from manual matrix operations? -

while reading code ucmerced's tripath toolkit, came across these

float xmin, xmax, ymin, ymax;  float mat[16] = { 1, 0, 0, 0,     0, 1, 0, 0,     0, 0, 1, 0,     0, 0, 0, 1 }; thelct->get_bounds ( xmin, xmax, ymin, ymax ); glmatrixmode ( gl_modelview ); glloadidentity ();  float width = xmax-xmin; float height = ymax-ymin; mat[0]=mat[5]=mat[10]= 1.8f * (1 / (width > height ? width : height)); glmultmatrixf ( mat ); mat[0]=mat[5]=mat[10]= 1;  mat[12]=-(xmin+w/2); mat[13]=-(ymin+h/2); glmultmatrixf ( mat ); 

in first transformation, first 3 diagonal 1's in matrix multiplied factor. limited knowledge of identity matrix, appears scaling factor.

the second transformation, however, don't understand:

mat[12]=-(xmin+w/2); mat[13]=-(ymin+h/2); glmultmatrixf ( mat ); 

first of all, don't know means change indices 12 , 13 in such matrix. i'm trying figure out reading wikipedia page on transformations, guess don't have enough math-related domain knowledge make sense of it.

whereas opengl resources can find don't seem modify matrices in manner, rather use functions glscalef.

how can relate manual matrix transformations such above scaling, shearing, translating, , rotating?

the first matrix, correctly guessed, uniform scale matrix. second matrix translation (along x , y axis). note (fixed function matrix stack of the) gl uses column major memory layout, translation part in m[12], m[13], m[14] (see answer 9.005 in old gl faq). combined transformation not perspective projection (that require (m[3], m[7], m[11]) not null vector), orthogonal one.

for easy explanation of how these numbers can geometrically interpreted, might find this article useful.