ÉÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ» º Mathematik Formules º º º º By Volpone of Malorean Effect for crazy mathematicians like Toto º º º º Bibliographie : -Fractint.doc (of Fractint 17.1) º º -Pour l'honneur de l'esprit humain º º (les math‚matiques aujourd'hui) º º Jean Dieudonn‚ º º º º Version 2.0 º º Release January 1994 º º º ÈÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍͼ --------------------------- Constantes Part --------------------------------- ã =3.14159265359 ã/2 =1.5707963268 2*ã =6.28318530718 e =2.71828182846 ----------------------- Complexe definitions Part ------------------------------- Z=x+i*y Ä Z=x-i*y Z=r*e^(i*à) r=û(x^2+y^2) Tan(à)=y/x Re(Z) = x Im(Z) = y ----------------------- Complexe formules Part ------------------------------- Ä Z^-1=Z/(³Z³^2) ³Z+Z'³<=³Z³+³Z'³ Ä Z*Z=³Z³^2 n n û(Z)= û(r)*e^i*(à/n) ³n ³ n ³ û(Z)³ = û³Z³ = (x^2+y^2)^(1/(2*n)) Z^n=1 -> Z=e^i*(à/n) ----------------------- Z1=û(Z2) Part ------------------------------- Z1=û(Z2) Re(Z1)=[û(³Z2³+X2)]/4 º Im(Z1)=[û(³Z2³-X2)]/4 Re(Z1)/Im(Z1)=(³Z2³+x2)/³y2³ Re(Z1)*Im(Z1)=³y2³/16 ------------------ Ln \ Log \ Exp \ Z^C Part ------------------------ * xîR ln(x+iy) = (1/2)ln(x*x + y*y) + i(arctan(y/x) + 2kPi) (k = 0, +-1, +-2, +-....) Ln(i*y) (y>0) = Ln(y)+i*ã/2 (y<0) = Ln(y)-i*ã/2 z^z = e^(log(z)*z) e^i*a = Cos(a) + i*Sin(a) e^(x+iy) = e^x * e^(i*y) = (Ch(x) + Sh(x)) * (cos(y) + i*sin(y)) = e^x * (cos(y) + i*sin(y)) = (e^x * cos(y)) + i(e^x * sin(y)) Z=x+i*y C=a+i*b * xîR ³Z³^a Z^C = ----------------- * e^i*(a*ArcTan(y/x) + b*Ln³Z³ ) e^(ArcTan(y/x)) -------------------- Circular real function Part -------------------------- xîR yîR Sh(x) = [e^x - e^(-x)]/2 Ch(x) = [e^x + e^(-x)]/2 Th(x) = Sh(x)+Ch(x) = e^x Sin (x+y) = Sin(x)*Cos(y)+Cos(x)*Sin(y) Sin (x-y) = Sin(x)*Cos(y)-Cos(x)*Sin(y) Cos (x+y) = Cos(x)*Cos(y)-Sin(x)*Sin(y) Cos (x-y) = Cos(x)*Cos(y)+Sin(x)*Sin(y) Tan (x+y) = [Tan(x)+Tan(b)] / [1-Tan(x)*Tan(y)] Tan (x-y) = [Tan(x)-Tan(b)] / [1+Tan(x)*Tan(y)] ATan(x+y) = ------------------- Circular complex function Part ------------------------ sin (x+iy) = sin(x)Ch(y) + icos(x)Sh(y) cos (x+iy) = cos(x)Ch(y) - isin(x)Sh(y) Sh(x+iy) = Sh(x)cos(y) + iCh(x)sin(y) Ch(x+iy) = Ch(x)cos(y) + iSh(x)sin(y) sin(2x) Sh(2y) tan(x+iy) = ------------------ + i------------------ cos(2x) + Ch(2y) cos(2x) + Ch(2y) Sh(2x) sin(2y) tanh(x+iy) = ------------------ + i------------------ Ch(2x) + cos(2y) Ch(2x) + cos(2y) sin(2x) - i*Sh(2y) cotan(x+iy) = -------------------- Ch(2y) - cos(2x) Sh(2x) - i*sin(2y) cotanh(x+iy) = -------------------- Ch(2x) - cos(2y) -------------------- Expoly1 Part ----------------------------- ³ The Expoly1 formule is use for the matematica 2 ³ ³ Expoly1 is create by Volpone Of Malorean Effect ³ ------------------------------------------------------------------ P(x)=A.x^2+B.X+C d P'(x) = --- P(x) = 2*A.x+B dx d P''(x) = --- P'(x) = 2*A dx d P'''(x) = --- P''(x) = 0 dx -The formule d Fx(0)=P(X) Fx(N) = Fx(N-1) + ----Fx(N-1) dx -Example :first number of suite (Fx(1)=P(x)+P'(x) ) d (Fx(2)=P(x)+P'(x) + ---[P(x)+P'(x)] dx =P(x)+P'(x)+[P'(x)+P''(x)] =P(x)+2*P'(x)+P''(x) - The formules N-1 Fx(N)=A.x^2 + (B+2*A*N).X + C+B*N+2*A*ä i i=1 N-1 Fx(N)=P(x) + N*P'(X) + ä i *P''(x) i=1 Fx(N+1)=Fx(N)+P'(x)+N*P''(x) Exemples : F(x,y) = Px(x) + Px(y) = Color to pixel(x,y) ---------------------------- Vector Part ------------------------------- -> ³Ux -> ³Vx U ³Uy V ³Vy ³Uz ³Vy -> -> º -> -> ³Uy*Vz-Vy*Uz U ù V = Ux*Vx+Uy*Vy+Uz*Vz º U ^ V = ³Uz*Vx-Vz*Ux º ³Ux*Vy-Vx*Uy -> -> -> -> -> -> -> -> -> U ^ (V ^ W ) = (U . W )*V - (U . V )*W ----------------------- Quaternion formules Part ------------------------------- Quarternions has discover by Hamilton in 1843 On pose Qu = ensembles nombres quarternions Q îQu -> Q1=A +i*B +j*C +k*D ÉÍÍÍÍÍÍÍÍÍÍËÍÍÍÍÍÍÍÍÍËÍÍÍÍÍÍÍÍËÍÍÍÍÍÍÍÍ» Q1îQu -> Q1=A1+i*B1+j*C1+k*D1 º i^2=-1 º i*j=+k º i*k=-j º j*k=+i º Q2îQu -> Q2=A2+i*B2+j*C2+k*D2 º k^2=-1 º j*i=-k º k*i=+j º k*j=-i º ÈÍÍÍÍÍÍÍÍÍÍÊÍÍÍÍÍÍÍÍÍÊÍÍÍÍÍÍÍÍÊÍÍÍÍÍÍÍͼ ³Q³=û(A^2+B^2+C^2+D^2) Q1+Q2=(A1+A2)+(B1+B2)*i+(C1+C2)*j+(D1+D2)*k Q1*Q2= (A1*A2-B1*B2-C1*C2-D1*D2) +(A1*B2+B1*A2+C1*D2+D1*C2)*i +(A1*C2-B1*D2+C1*A2+D1*B2)*j +(A1*D2+B1*C2-C1*B2+D1*D3)*k ZORG !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!