Concrete expressions of circles on Blum cyclides

Lemma

A parameter expression of a circle

x:x0+((2r(U1+V1 t))/((((U1+V1 t))^2)+(((U2+V2 t))^2)+(((U3+V3 t))^2)))

y:y0+((2r(U2+V2 t))/((((U1+V1 t))^2)+(((U2+V2 t))^2)+(((U3+V3 t))^2)))

z:z0+((2r(U3+V3 t))/((((U1+V1 t))^2)+(((U2+V2 t))^2)+(((U3+V3 t))^2)))

contained in  y-y0=a(x-x0)+b(z-z0)
 with center (x0+W1,y0+W2,z0+W3) and radius (Sqrt[W1^2+W2^2+W3^2]). We find a, b, W1, W2, W3.

Simplify[Cancel[((((U1+V1 t))^2)+(((U2+V2 t))^2)+(((U3+V3 t))^2))((((2r(U1+V1 t))/((((U1+V1 t))^2)+(((U2+V2 t))^2)+(((U3+V3 t))^2)))-W1)^2+(((2r(U2+V2 t))/((((U1+V1 t))^2)+(((U2+V2 t))^2)+(((U3+V3 t))^2)))-W2)^2+(((2r(U3+V3 t))/((((U1+V1 t))^2)+(((U2+V2 t))^2)+(((U3+V3 t))^2)))-W3)^2-W1^2-W2^2-W3^2)]]

Solve[{V2-a V1-b V3==0,U2-a U1-b U3==0},{a,b}]

Solve[{V1 W1+V2 W2+V3 W3==0,U1 W1+U2 W2+U3 W3-r==0,W2-(-((-U3 V2+U2 V3)/(U3 V1-U1 V3))) W1-(-((U2 V1-U1 V2)/(-U3 V1+U1 V3))) W3==0},{W1,W2,W3}]

Therefore W1^2+W2^2+W3^2 is

Simplify[(-r(U2 V1 V2-U1 (V2^2)+U3 V1 V3-U1 (V3^2))/((U2^2) (V1^2)+(U3^2) (V1^2)-2 U1 U2 V1 V2+(U1^2) (V2^2)+(U3^2) (V2^2)-2 U1 U3 V1 V3-2 U2 U3 V2 V3+(U1^2) (V3^2)+(U2^2) (V3^2)))^2+(-r(-U2 (V1^2)+U1 V1 V2+U3 V2 V3-U2 (V3^2))/((U2^2) (V1^2)+(U3^2) (V1^2)-2 U1 U2 V1 V2+(U1^2) (V2^2)+(U3^2) (V2^2)-2 U1 U3 V1 V3-2 U2 U3 V2 V3+(U1^2) (V3^2)+(U2^2) (V3^2)))^2+(-r(-U3 (V1^2)-U3 (V2^2)+U1 V1 V3+U2 V2 V3)/((U2^2) (V1^2)+(U3^2) (V1^2)-2 U1 U2 V1 V2+(U1^2) (V2^2)+(U3^2) (V2^2)-2 U1 U3 V1 V3-2 U2 U3 V2 V3+(U1^2) (V3^2)+(U2^2) (V3^2)))^2]

Proposition(circles on a Blum surface )

Let (x0,y0,z0) be any point on a Blum cyclide{(x^2+y^2+z^2)^2-2 a x^2-2 b y^2-2 c z^2+d^2=0}. Here we suppose  a>c>d>0, -b>d. 
We consider the inversion at (x0,y0,z0) as follows(x1,y1,z1 are new variables):

Simplify[Cancel[(x1^2+y1^2+z1^2)^2((x^2+y^2+z^2)^2-2 a x^2-2 b y^2-2 c z^2+d^2)-16((A x0 x1+B y0 y1+C0 z0 z1)(x1^2+y1^2+z1^2)+A x1^2+B y1^2+C0 z1^2+(x0 x1+y0 y1+z0 z1+1)^2)/.{x->x0+2 x1/(x1^2+y1^2+z1^2),y->y0+2 y1/(x1^2+y1^2+z1^2),z->z0+2 z1/(x1^2+y1^2+z1^2),A->(x0^2+y0^2+z0^2-a)/2,B->(x0^2+y0^2+z0^2-b)/2,C0->(x0^2+y0^2+z0^2-c)/2,d->√(2 a x0^2+2 b y0^2+2 c z0^2-(x0^2+y0^2+z0^2)^2)}]]

Therefore Blum equation 
(x^2+y^2+z^2)^2-2 a x^2-2 b y^2-2 c z^2+d^2=0  
reduces to 
(A x0 x1+B y0 y1+C0 z0 z1)(x1^2+y1^2+z1^2)+A x1^2+B y1^2+C0 z1^2+(x0 x1+y0 y1+z0 z1+1)^2=0 under the inversion at (x0,y0,z0).

We impose an aditional condition 
A x0 x1+B y0 y1+C0 z0 z1+k=0 with a parameter k, and  we eliminate z1:

Collect[C0^2z0^2(-k(x1^2+y1^2+z1^2)+A x1^2+B y1^2+C0 z1^2+(x0 x1+y0 y1+z0 z1+1)^2)/.{z1->-(A x0 x1+B y0 y1+k)/(C0 z0)},{x1,y1},Simplify]

The coefficient of x1^2: A2

Simplify[A C0 (C0-2 (x0^2)) (z0^2)+(C0^2) (-k+(x0^2)) (z0^2)+(A^2) (x0^2) (C0-k+(z0^2))-(C0^2 z0^2(A-k)+A^2 x0^2(C0-k)+z0^2 x0^2(C0-A)^2)]

A2:C0^2 z0^2(A-k)+A^2 x0^2(C0-k)+z0^2 x0^2(C0-A)^2

The coefficient of x1 y1: B2

Simplify[2 x0 y0  (C0 (-B+C0) (z0^2)+A (-C0 (z0^2)+B (C0-k+(z0^2))))-(2 A B(C0-k)x0 y0+2(C0-A)(C0-B)x0 y0 z0^2)]

B2: 2 A B(C0-k)x0 y0+2(C0-A)(C0-B)x0 y0 z0^2

The coefficient of y1^2: C2

Simplify[B C0 (C0-2 (y0^2)) (z0^2)+(C0^2) (-k+(y0^2)) (z0^2)+(B^2) (y0^2) (C0-k+(z0^2))-(C0^2 z0^2(B-k)+B^2 y0^2(C0-k)+z0^2 y0^2(C0-B)^2)]

C2: C0^2 z0^2(B-k)+B^2 y0^2(C0-k)+z0^2 y0^2(C0-B)^2

The coefficient of x1: D2

Simplify[2 (C0-k) x0 (C0 (z0^2)+A (k-(z0^2)))-(2(C0-k)k A x0+2(C0-A)(C0-k)x0 z0^2)]

D2: 2(C0-k)k A x0+2(C0-A)(C0-k)x0 z0^2

The coefficient of y1: E2

Simplify[2 (C0-k) y0(C0 (z0^2)+B (k-(z0^2)))-(2(C0-k)k B y0+2(C0-B)(C0-k)y0 z0^2)]

E2: 2(C0-k)k B y0+2(C0-B)(C0-k)y0 z0^2

The constant term: F2

Simplify[(C0^2) (z0^2)+C0 k (k-2 (z0^2))+(k^2) (-k+(z0^2))-((C0-k)k^2+(C0-k)^2 z0^2)]

F2:(C0-k)k^2+(C0-k)^2 z0^2

Therefore  for the splitting of A2 x1^2+B2 x1 y1 +C2 y1^2+D2 x1+E2 y1+F2 into two first order polynomials, we consider the discriminant in x1 of
 A2 x1^2+B2 x1 y1 +C2 y1^2+D2 x1+E2 y1+F2=A2 x1^2+(B2 y1+D2)x1 +C2 y1^2+E2 y1+F2

Collect[(B2 y1+D2)^2-4 A2 (C2 y1^2+E2 y1+F2),y1]

Hence  for the splitting, k must satisfies the vanishing of the discriminant of the above quadratic polynomial in y1, and the positivity of (B2^2)-4 A2 C2.

As for the vanishing of  the discriminant in y1:

Simplify[(B2 D2-2 A2 E2)^2-(B2^2-4 A2 C2)(D2^2-4 A2 F2)]

Hence we impose -B2 D2 E2+A2 (E2^2)+(B2^2) F2+C2 ((D2^2)-4 A2 F2)=0 on k:

Simplify[-B2 D2 E2+A2 (E2^2)+(B2^2) F2+C2 ((D2^2)-4 A2 F2)/.{A2->C0^2 z0^2(A-k)+A^2 x0^2(C0-k)+z0^2 x0^2(C0-A)^2,B2->2 A B(C0-k)x0 y0+2(C0-A)(C0-B)x0 y0 z0^2,C2->C0^2 z0^2(B-k)+B^2 y0^2(C0-k)+z0^2 y0^2(C0-B)^2,D2->2(C0-k)k A x0+2(C0-A)(C0-k)x0 z0^2,E2-> 2(C0-k)k B y0+2(C0-B)(C0-k)y0 z0^2,F2->(C0-k)k^2+(C0-k)^2 z0^2}]

The last factor is equal to

(k+(1/2) (-d-(x0^2)-(y0^2)-(z0^2))) (k+(1/2) (d-(x0^2)-(y0^2)-(z0^2)))

because

Simplify[(x0^2+y0^2+z0^2)^2-d^2-4(A x0^2+B y0^2+C0 z0^2)/.{A->(x0^2+y0^2+z0^2-a)/2,B->(x0^2+y0^2+z0^2-b)/2,C0->(x0^2+y0^2+z0^2-c)/2,d->√(2 a x0^2+2 b y0^2+2 c z0^2-(x0^2+y0^2+z0^2)^2)}]

Thus k should be either one of  A, B, C0, (x0^2+y0^2+z0^2-d)/2,(x0^2+y0^2+z0^2+d)/2

Further concerning the positivity of B2^2-4 A2 C2, for each k we have the following:

Case k=A=(x0^2+y0^2+z0^2-a)/2

Simplify[(B2^2-4 A2 C2/.{A2->C0^2 z0^2(A-k)+A^2 x0^2(C0-k)+z0^2 x0^2(C0-A)^2,B2->2 A B(C0-k)x0 y0+2(C0-A)(C0-B)x0 y0 z0^2,C2->C0^2 z0^2(B-k)+B^2 y0^2(C0-k)+z0^2 y0^2(C0-B)^2})/.{k->A}]

Simplify[(A^2)+B (y0^2)+C0 (z0^2)-A ((y0^2)+(z0^2))/.{A->(x0^2+y0^2+z0^2-a)/2,B->(x0^2+y0^2+z0^2-b)/2,C0->(x0^2+y0^2+z0^2-c)/2}]

This is equal to ((x0^2+y0^2+z0^2)^2-2 a x0^2-2 b y0^2-2 c z0^2+a^2)/4=(a^2-d^2)/4

Hence the last factor is (a^2-d^2)/4. Therefore

(a-b)(c-a)(a+d)(a-d)>0

Case k=B=(x0^2+y0^2+z0^2-b)/2

Simplify[(B2^2-4 A2 C2/.{A2->C0^2 z0^2(A-k)+A^2 x0^2(C0-k)+z0^2 x0^2(C0-A)^2,B2->2 A B(C0-k)x0 y0+2(C0-A)(C0-B)x0 y0 z0^2,C2->C0^2 z0^2(B-k)+B^2 y0^2(C0-k)+z0^2 y0^2(C0-B)^2})/.{k->B}]

Simplify[(B^2)+A (x0^2)+C0 (z0^2)-B ((x0^2)+(z0^2))/.{A->(x0^2+y0^2+z0^2-a)/2,B->(x0^2+y0^2+z0^2-b)/2,C0->(x0^2+y0^2+z0^2-c)/2}]

This is equal to (b^2-d^2)/4. Therefore
(a-b)(b-c)(b+d)(b-d)>0

Case k=C0=(x0^2+y0^2+z0^2-c)/2

Simplify[(B2^2-4 A2 C2/.{A2->C0^2 z0^2(A-k)+A^2 x0^2(C0-k)+z0^2 x0^2(C0-A)^2,B2->2 A B(C0-k)x0 y0+2(C0-A)(C0-B)x0 y0 z0^2,C2->C0^2 z0^2(B-k)+B^2 y0^2(C0-k)+z0^2 y0^2(C0-B)^2})/.{k->C0}]

Simplify[(C0^2)+A (x0^2)+B (y0^2)-C0 ((x0^2)+(y0^2))/.{A->(x0^2+y0^2+z0^2-a)/2,B->(x0^2+y0^2+z0^2-b)/2,C0->(x0^2+y0^2+z0^2-c)/2}]

This is equal to (c^2-d^2)/4. Therefore
(a-c)(b-c)(c+d)(d-c)>0

Case k=(x0^2+y0^2+z0^2-d)/2

Simplify[((B2^2-4 A2 C2)-(C0^2z0^2(d-a)(d-b)(d-c)(x0^2+y0^2+z0^2+d)^2/8)/.{A2->C0^2 z0^2(A-k)+A^2 x0^2(C0-k)+z0^2 x0^2(C0-A)^2,B2->2 A B(C0-k)x0 y0+2(C0-A)(C0-B)x0 y0 z0^2,C2->C0^2 z0^2(B-k)+B^2 y0^2(C0-k)+z0^2 y0^2(C0-B)^2})/.{k->(x0^2+y0^2+z0^2-d)/2,A->(x0^2+y0^2+z0^2-a)/2,B->(x0^2+y0^2+z0^2-b)/2,C0->(x0^2+y0^2+z0^2-c)/2}/.d->√(2 a x0^2+2 b y0^2+2 c z0^2-(x0^2+y0^2+z0^2)^2)]

Therefore
(d-a)(d-b)(d-c)>0

Case k=(x0^2+y0^2+z0^2+d)/2

Simplify[((B2^2-4 A2 C2)-(-C0^2z0^2(d+a)(d+b)(d+c)(x0^2+y0^2+z0^2-d)^2/8)/.{A2->C0^2 z0^2(A-k)+A^2 x0^2(C0-k)+z0^2 x0^2(C0-A)^2,B2->2 A B(C0-k)x0 y0+2(C0-A)(C0-B)x0 y0 z0^2,C2->C0^2 z0^2(B-k)+B^2 y0^2(C0-k)+z0^2 y0^2(C0-B)^2})/.{k->(x0^2+y0^2+z0^2+d)/2,A->(x0^2+y0^2+z0^2-a)/2,B->(x0^2+y0^2+z0^2-b)/2,C0->(x0^2+y0^2+z0^2-c)/2}/.d->√(2 a x0^2+2 b y0^2+2 c z0^2-(x0^2+y0^2+z0^2)^2)]

Therefore
-(d+a)(d+b)(d+c)>0

Hence, only  k=(x0^2+y0^2+z0^2-c)/2, (x0^2+y0^2+z0^2-d)/2,(x0^2+y0^2+z0^2+d)/2 give the positive signatures.

Case 1. k=C0=(x0^2+y0^2+z0^2-c)/2

Solve the system of equations
 with respect to y1,z1:
-k(x1^2+y1^2+z1^2)+A x1^2+B y1^2+C0 z1^2+(x0 x1+y0 y1+z0 z1+1)^2==0, A x0 x1+B y0 y1+C0 z0 z1+k==0.
Setting v1=(Sqrt[(a-c) (c-b)((c^2)-(d^2))]), we show that the following is the solution:

y1->(2 (a-c) (-b+c) x0 y0-v1(-c+(x0^2)+(y0^2)+(z0^2)) )x1/((c-b)((c^2)-(d^2)+2 (a-c)  (x0^2))), z1->((c-b)  (d^2-(c^2)+2 (c-a) (x0^2))+　((b-c) x0  (( a-c) (x0^2+y0^2-z0^2)+a c-d^2)+v1 y0((x0^2) +(y0^2) +(z0^2)-b)) x1 )/((c-b)z0  ((c^2)-(d^2)+2 (a-c)  (x0^2)))

Simplify[(A x0 x1+B y0 y1+C0 z0 z1+k/.{A->(x0^2+y0^2+z0^2-a)/2,B->(x0^2+y0^2+z0^2-b)/2,C0->(x0^2+y0^2+z0^2-c)/2,k->(x0^2+y0^2+z0^2-c)/2})/.{y1->(2 (a-c) (-b+c) x0 y0-v1(-c+(x0^2)+(y0^2)+(z0^2)) )x1/((c-b)((c^2)-(d^2)+2 (a-c)  (x0^2))), z1->((c-b)  (d^2-(c^2)+2 (c-a) (x0^2))+　((b-c) x0  (( a-c) (x0^2+y0^2-z0^2)+a c-d^2)+v1 y0((x0^2) +(y0^2) +(z0^2)-b)) x1 )/((c-b)z0  ((c^2)-(d^2)+2 (a-c)  (x0^2)))}/.{d->√(2 a x0^2+2 b y0^2+2 c z0^2-(x0^2+y0^2+z0^2)^2)}]

Simplify[(-k(x1^2+y1^2+z1^2)+A x1^2+B y1^2+C0 z1^2+(x0 x1+y0 y1+z0 z1+1)^2/.{A->(x0^2+y0^2+z0^2-a)/2,B->(x0^2+y0^2+z0^2-b)/2,C0->(x0^2+y0^2+z0^2-c)/2,k->(x0^2+y0^2+z0^2-c)/2})/.{y1->(2 (a-c) (-b+c) x0 y0-v1(-c+(x0^2)+(y0^2)+(z0^2)) )x1/((c-b)((c^2)-(d^2)+2 (a-c)  (x0^2))), z1->((c-b)  (d^2-(c^2)+2 (c-a) (x0^2))+　((b-c) x0  (( a-c) (x0^2+y0^2-z0^2)+a c-d^2)+v1 y0((x0^2) +(y0^2) +(z0^2)-b)) x1 )/((c-b)z0  ((c^2)-(d^2)+2 (a-c)  (x0^2)))}/.{v1->(Sqrt[(a-c) (c-b)((c^2)-(d^2))])}/.{d->√(2 a x0^2+2 b y0^2+2 c z0^2-(x0^2+y0^2+z0^2)^2)}]

The original coordinate (x,y,z) of the point (x1,y1,z1) is

x :

Simplify[x0+2 x1/(x1^2+y1^2+z1^2)/.{y1->(2 (a-c) (-b+c) x0 y0-v1(-c+(x0^2)+(y0^2)+(z0^2)) )x1/((c-b)((c^2)-(d^2)+2 (a-c)  (x0^2))),z1->((c-b)  (d^2-(c^2)+2 (c-a) (x0^2))+　((b-c) x0  (( a-c) (x0^2+y0^2-z0^2)+a c-d^2)+v1 y0((x0^2) +(y0^2) +(z0^2)-b)) x1 )/((c-b)z0  ((c^2)-(d^2)+2 (a-c)  (x0^2)))}]

Simplify[(2 x1)/((x1^2)+(((x1^2) (((-2 (a-c) (-b+c) x0 y0+v1 (-c+(x0^2)+(y0^2)+(z0^2))))^2))/((((b-c))^2) ((((c^2)-(d^2)+2 (a-c) (x0^2)))^2)))+((((-b+c) (-(c^2)+(d^2)+2 (-a+c) (x0^2))+x1 (v1 y0 (-b+(x0^2)+(y0^2)+(z0^2))+(b-c) x0 (a c-(d^2)+(a-c) ((x0^2)+(y0^2)-(z0^2))))))^2)/((((b-c))^2) ((((c^2)-(d^2)+2 (a-c) (x0^2)))^2) (z0^2)))-2(B1^2)(z0^2)x1/((B1^2)(z0^2)(((x1))^2)+(B2^2) (z0^2)(((x1))^2)+(((B1+B3 x1))^2))/.{B1->(b-c) ((c^2)-(d^2)+2 (a-c) (x0^2)),B2->2 (a-c) (-b+c) x0 y0-v1 (-c+(x0^2)+(y0^2)+(z0^2)),B3->v1 y0 (-b+(x0^2)+(y0^2)+(z0^2))+(b-c) x0 (a c-(d^2)+(a-c) ((x0^2)+(y0^2)-(z0^2)))}]

y:

Simplify[y0+2 y1/(x1^2+y1^2+z1^2)/.{y1->(2 (a-c) (-b+c) x0 y0-v1(-c+(x0^2)+(y0^2)+(z0^2)) )x1/((c-b)((c^2)-(d^2)+2 (a-c)  (x0^2))),z1->((c-b)  (d^2-(c^2)+2 (c-a) (x0^2))+　((b-c) x0  (( a-c) (x0^2+y0^2-z0^2)+a c-d^2)+v1 y0((x0^2) +(y0^2) +(z0^2)-b)) x1 )/((c-b)z0  ((c^2)-(d^2)+2 (a-c)  (x0^2)))}]

Simplify[(2 x1 (2 (a-c) (-b+c) x0 y0-v1 (-c+(x0^2)+(y0^2)+(z0^2))))/((-b+c) ((c^2)-(d^2)+2 (a-c) (x0^2)) ((x1^2)+(((x1^2) (((-2 (a-c) (-b+c) x0 y0+v1 (-c+(x0^2)+(y0^2)+(z0^2))))^2))/((((b-c))^2) ((((c^2)-(d^2)+2 (a-c) (x0^2)))^2)))+((((-b+c) (-(c^2)+(d^2)+2 (-a+c) (x0^2))+x1 (v1 y0 (-b+(x0^2)+(y0^2)+(z0^2))+(b-c) x0 (a c-(d^2)+(a-c) ((x0^2)+(y0^2)-(z0^2))))))^2)/((((b-c))^2) ((((c^2)-(d^2)+2 (a-c) (x0^2)))^2) (z0^2))))-(-2 B1 B2 (z0^2)x1/((B1^2)(z0^2)(((x1))^2)+(B2^2) (z0^2)(((x1))^2)+(((B1+B3 x1))^2)))/.{B1->(b-c) ((c^2)-(d^2)+2 (a-c) (x0^2)),B2->2 (a-c) (-b+c) x0 y0-v1 (-c+(x0^2)+(y0^2)+(z0^2)),B3->v1 y0 (-b+(x0^2)+(y0^2)+(z0^2))+(b-c) x0 (a c-(d^2)+(a-c) ((x0^2)+(y0^2)-(z0^2)))}]

z:

Simplify[z0+2 z1/(x1^2+y1^2+z1^2)/.{y1->(2 (a-c) (-b+c) x0 y0-v1(-c+(x0^2)+(y0^2)+(z0^2)) )x1/((c-b)((c^2)-(d^2)+2 (a-c)  (x0^2))),z1->((c-b)  (d^2-(c^2)+2 (c-a) (x0^2))+　((b-c) x0  (( a-c) (x0^2+y0^2-z0^2)+a c-d^2)+v1 y0((x0^2) +(y0^2) +(z0^2)-b)) x1 )/((c-b)z0  ((c^2)-(d^2)+2 (a-c)  (x0^2)))}]

Simplify[(2 ((-b+c) (-(c^2)+(d^2)+2 (-a+c) (x0^2))+x1 (v1 y0 (-b+(x0^2)+(y0^2)+(z0^2))+(b-c) x0 (a c-(d^2)+(a-c) ((x0^2)+(y0^2)-(z0^2))))))/((-b+c) ((c^2)-(d^2)+2 (a-c) (x0^2)) z0 ((x1^2)+(((x1^2) (((-2 (a-c) (-b+c) x0 y0+v1 (-c+(x0^2)+(y0^2)+(z0^2))))^2))/((((b-c))^2) ((((c^2)-(d^2)+2 (a-c) (x0^2)))^2)))+((((-b+c) (-(c^2)+(d^2)+2 (-a+c) (x0^2))+x1 (v1 y0 (-b+(x0^2)+(y0^2)+(z0^2))+(b-c) x0 (a c-(d^2)+(a-c) ((x0^2)+(y0^2)-(z0^2))))))^2)/((((b-c))^2) ((((c^2)-(d^2)+2 (a-c) (x0^2)))^2) (z0^2))))-(-2 B1 z0(B1+B3 x1)/((B1^2)(z0^2)(((x1))^2)+(B2^2) (z0^2)(((x1))^2)+(((B1+B3 x1))^2)))/.{B1->(b-c) ((c^2)-(d^2)+2 (a-c) (x0^2)),B2->2 (a-c) (-b+c) x0 y0-v1 (-c+(x0^2)+(y0^2)+(z0^2)),B3->v1 y0 (-b+(x0^2)+(y0^2)+(z0^2))+(b-c) x0 (a c-(d^2)+(a-c) ((x0^2)+(y0^2)-(z0^2)))}]

Therefore

x: x0+2(B1^2)(z0^2)x1/((B1^2)(z0^2)(((x1))^2)+(B2^2) (z0^2)(((x1))^2)+(((B1+B3 x1))^2))

y:y0-2 B1 B2 (z0^2)x1/((B1^2)(z0^2)(((x1))^2)+(B2^2) (z0^2)(((x1))^2)+(((B1+B3 x1))^2))

z:z0-2 B1 z0(B1+B3 x1)/((B1^2)(z0^2)(((x1))^2)+(B2^2) (z0^2)(((x1))^2)+(((B1+B3 x1))^2))

where

B1->(b-c) ((c^2)-(d^2)+2 (a-c) (x0^2))

B2->2 (a-c) (-b+c) x0 y0-v1 (-c+(x0^2)+(y0^2)+(z0^2))

B3->v1 y0 (-b+(x0^2)+(y0^2)+(z0^2))+(b-c) x0 (a c-(d^2)+(a-c) ((x0^2)+(y0^2)-(z0^2)))

Hence by setting
r->z0 B1, U1->0,U2->0, U3->-B1,V1->z0 B1, V2->-z0 B2,V3->-B3, we can apply Lemma.

Thus  t is

Simplify[-((-U3 V2+U2 V3)/(U3 V1-U1 V3))/.{U1->0,U2->0, U3->-B1,V1->z0 B1, V2->-z0 B2,V3->-B3}]

s is

Simplify[-((U2 V1-U1 V2)/(-U3 V1+U1 V3))/.{U1->0,U2->0, U3->-B1,V1->z0 B1, V2->-z0 B2,V3->-B3}]

Consequently

T=t=-(B2/B1)=-((2 (a-c) (-b+c) x0 y0-v1 (-c+(x0^2)+(y0^2)+(z0^2)))/((b-c) ((c^2)-(d^2)+2 (a-c) (x0^2))))

Here z0 is

Solve[{(x0^2+y0^2+z0^2)^2-2 a x0^2-2 b y0^2-2 c z0^2+d^2==0},z0]

Hence  -c+(x0^2)+(y0^2)+(z0^2) is

(Sqrt[(c^2)-(d^2)+2 (a-c) (x0^2)+2( b-c) (y0^2)]),-(Sqrt[(c^2)-(d^2)+2 (a-c) (x0^2)+2( b-c) (y0^2)])

The center of the circle is

x:

Simplify[x0+(r (-U2 V1 V2+U1 (V2^2)-U3 V1 V3+U1 (V3^2)))/((U2^2) (V1^2)+(U3^2) (V1^2)-2 U1 U2 V1 V2+(U1^2) (V2^2)+(U3^2) (V2^2)-2 U1 U3 V1 V3-2 U2 U3 V2 V3+(U1^2) (V3^2)+(U2^2) (V3^2))/.{r->z0 B1,U1->0,U2->0, U3->-B1,V1->z0 B1, V2->-z0 B2,V3->-B3}]

y:

Simplify[y0+(r (U2 (V1^2)-U1 V1 V2-U3 V2 V3+U2 (V3^2)))/((U2^2) (V1^2)+(U3^2) (V1^2)-2 U1 U2 V1 V2+(U1^2) (V2^2)+(U3^2) (V2^2)-2 U1 U3 V1 V3-2 U2 U3 V2 V3+(U1^2) (V3^2)+(U2^2) (V3^2))/.{r->z0 B1,U1->0,U2->0, U3->-B1,V1->z0 B1, V2->-z0 B2,V3->-B3}]

z:

Simplify[z0+(r (U3 (V1^2)+U3 (V2^2)-U1 V1 V3-U2 V2 V3))/((U2^2) (V1^2)+(U3^2) (V1^2)-2 U1 U2 V1 V2+(U1^2) (V2^2)+(U3^2) (V2^2)-2 U1 U3 V1 V3-2 U2 U3 V2 V3+(U1^2) (V3^2)+(U2^2) (V3^2))/.{r->z0 B1,U1->0,U2->0, U3->-B1,V1->z0 B1, V2->-z0 B2,V3->-B3}]

(The radius)^2:

Simplify[((r^2) ((V1^2)+(V2^2)+(V3^2)))/((U3^2) ((V1^2)+(V2^2))-2 U1 U3 V1 V3-2 U2 V2 (U1 V1+U3 V3)+(U2^2) ((V1^2)+(V3^2))+(U1^2) ((V2^2)+(V3^2)))/.{r->z0 B1,U1->0,U2->0, U3->-B1,V1->z0 B1, V2->-z0 B2,V3->-B3}]

Case k=(x0^2+y0^2+z0^2-d)/2

Solve the system of equations
 with respect to y1,z1:
-k(x1^2+y1^2+z1^2)+A x1^2+B y1^2+C0 z1^2+(x0 x1+y0 y1+z0 z1+1)^2==0, A x0 x1+B y0 y1+C0 z0 z1+k==0.
Setting v2=(Sqrt[2(-b+d) (c-d)  (a-d)])

we show that the following is the solution:

y1->(2(c-d) y0 (b d-d^2+2 (a-b) x0^2+(b -d)((x0^2)+(y0^2)+(z0^2)))+2x0 y0 ((a b-d^2)(c -d )+(a-b)( c-d)(x0^2)-(a-b)( c-d)(y0^2)+((a+b)(c+d)-2 (a b+ c d))(z0^2))x1-v2 z0 (x0^2+y0^2+z0^2-c)((d +(x0^2)+(y0^2)+(z0^2)) x1+2 x0))/(2((y0^2)(d-c )((b^2)-(d^2)+2 (x0^2) (a-b))+(z0^2)(b-d)((d^2) -(c^2)+2(c-a)(x0^2) ))),z1->(-2 z0((b-d) ((d^2)-c d-(2 a-c-d) (x0^2)+(d-c) ((y0^2)+(z0^2)))-((a c-d^2)(b-d)+(b-d)(a-c)(x0^2-z0^2)+( b c-2 b d+d c-2 a c+a b+a d)y0^2)x0  x1)+v2 y0  (-b+(x0^2)+(y0^2)+(z0^2))  (2 x0+(d+(x0^2)+(y0^2)+(z0^2)) x1))/(2 ((-c+d) ((b^2)-(d^2)+2 (a-b) (x0^2)) (y0^2)+(b-d) (-(c^2)+(d^2)+2 (-a+c) (x0^2)) (z0^2)))

Simplify[(A x0 x1+B y0 y1+C0 z0 z1+k/.{A->(x0^2+y0^2+z0^2-a)/2,B->(x0^2+y0^2+z0^2-b)/2,C0->(x0^2+y0^2+z0^2-c)/2,k->(x0^2+y0^2+z0^2-d)/2})/.{y1->(2(c-d) y0 (b d-d^2+2 (a-b) x0^2+(b -d)((x0^2)+(y0^2)+(z0^2)))+2x0 y0 ((a b-d^2)(c -d )+(a-b)( c-d)(x0^2)-(a-b)( c-d)(y0^2)+((a+b)(c+d)-2 (a b+ c d))(z0^2))x1-v2 z0 (x0^2+y0^2+z0^2-c)((d +(x0^2)+(y0^2)+(z0^2)) x1+2 x0))/(2((y0^2)(d-c )((b^2)-(d^2)+2 (x0^2) (a-b))+(z0^2)(b-d)((d^2) -(c^2)+2(c-a)(x0^2) ))),z1->(-2 z0((b-d) ((d^2)-c d-(2 a-c-d) (x0^2)+(d-c) ((y0^2)+(z0^2)))-((a c-d^2)(b-d)+(b-d)(a-c)(x0^2-z0^2)+( b c-2 b d+d c-2 a c+a b+a d)y0^2)x0  x1)+v2 y0  (-b+(x0^2)+(y0^2)+(z0^2))  (2 x0+(d+(x0^2)+(y0^2)+(z0^2)) x1))/(2 ((-c+d) ((b^2)-(d^2)+2 (a-b) (x0^2)) (y0^2)+(b-d) (-(c^2)+(d^2)+2 (-a+c) (x0^2)) (z0^2)))}/.{d->√(2 a x0^2+2 b y0^2+2 c z0^2-(x0^2+y0^2+z0^2)^2)}]

Simplify[(-k(x1^2+y1^2+z1^2)+A x1^2+B y1^2+C0 z1^2+(x0 x1+y0 y1+z0 z1+1)^2/.{A->(x0^2+y0^2+z0^2-a)/2,B->(x0^2+y0^2+z0^2-b)/2,C0->(x0^2+y0^2+z0^2-c)/2,k->(x0^2+y0^2+z0^2-d)/2})/.{y1->(2(c-d) y0 (b d-d^2+2 (a-b) x0^2+(b -d)((x0^2)+(y0^2)+(z0^2)))+2x0 y0 ((a b-d^2)(c -d )+(a-b)( c-d)(x0^2)-(a-b)( c-d)(y0^2)+((a+b)(c+d)-2 (a b+ c d))(z0^2))x1-v2 z0 (x0^2+y0^2+z0^2-c)((d +(x0^2)+(y0^2)+(z0^2)) x1+2 x0))/(2((y0^2)(d-c )((b^2)-(d^2)+2 (x0^2) (a-b))+(z0^2)(b-d)((d^2) -(c^2)+2(c-a)(x0^2) ))),z1->(-2 z0((b-d) ((d^2)-c d-(2 a-c-d) (x0^2)+(d-c) ((y0^2)+(z0^2)))-((a c-d^2)(b-d)+(b-d)(a-c)(x0^2-z0^2)+( b c-2 b d+d c-2 a c+a b+a d)y0^2)x0  x1)+v2 y0  (-b+(x0^2)+(y0^2)+(z0^2))  (2 x0+(d+(x0^2)+(y0^2)+(z0^2)) x1))/(2 ((-c+d) ((b^2)-(d^2)+2 (a-b) (x0^2)) (y0^2)+(b-d) (-(c^2)+(d^2)+2 (-a+c) (x0^2)) (z0^2)))}/.{v2->(Sqrt[2(-b+d) (c-d)  (a-d)])}/.{d->√(2 a x0^2+2 b y0^2+2 c z0^2-(x0^2+y0^2+z0^2)^2)}]

We introduce C1,C2,C3,C4,C5 as
C1->(-c+d) ((b^2)-(d^2)+2 (a-b) (x0^2)) (y0^2)+(b-d) (-(c^2)+(d^2)+2 (-a+c) (x0^2)) (z0^2),C2->-v2 x0 z0 (-c+(x0^2)+(y0^2)+(z0^2))+(c-d) y0 (b d-(d^2)+2 (a-b) (x0^2)+(b-d) ((x0^2)+(y0^2)+(z0^2))),C3->-v2 z0 (-c+(x0^2)+(y0^2)+(z0^2)) (d+(x0^2)+(y0^2)+(z0^2))+2 x0 y0 ((c-d) (a b-(d^2))+(a-b) (c-d) (x0^2)-(a-b) (c-d) (y0^2)+((a+b) (c+d)-2 (a b+c d)) (z0^2)),C4->v2 x0 y0 (-b+(x0^2)+(y0^2)+(z0^2))+(-b+d) z0 ((d^2)-2 a (x0^2)-c (d-(x0^2)+(y0^2)+(z0^2))+d ((x0^2)+(y0^2)+(z0^2))),C5->v2 y0 (-b+(x0^2)+(y0^2)+(z0^2)) (d+(x0^2)+(y0^2)+(z0^2))+2 x0 z0 ((b-d) (a c-(d^2))+(b (c-2 d)+c d+a (b-2 c+d)) (y0^2)+(a-c) (b-d) ((x0^2)-(z0^2)))

Then, the original coordinate (x,y,z) of the point (x1,y1,z1) is

x:

Simplify[x0+2 x1/(x1^2+y1^2+z1^2)/.{y1->(2(c-d) y0 (b d-d^2+2 (a-b) x0^2+(b -d)((x0^2)+(y0^2)+(z0^2)))+2x0 y0 ((a b-d^2)(c -d )+(a-b)( c-d)(x0^2)-(a-b)( c-d)(y0^2)+((a+b)(c+d)-2 (a b+ c d))(z0^2))x1-v2 z0 (x0^2+y0^2+z0^2-c)((d +(x0^2)+(y0^2)+(z0^2)) x1+2 x0))/(2((y0^2)(d-c )((b^2)-(d^2)+2 (x0^2) (a-b))+(z0^2)(b-d)((d^2) -(c^2)+2(c-a)(x0^2) ))),z1->(-2 z0((b-d) ((d^2)-c d-(2 a-c-d) (x0^2)+(d-c) ((y0^2)+(z0^2)))-((a c-d^2)(b-d)+(b-d)(a-c)(x0^2-z0^2)+( b c-2 b d+d c-2 a c+a b+a d)y0^2)x0  x1)+v2 y0  (-b+(x0^2)+(y0^2)+(z0^2))  (2 x0+(d+(x0^2)+(y0^2)+(z0^2)) x1))/(2 ((-c+d) ((b^2)-(d^2)+2 (a-b) (x0^2)) (y0^2)+(b-d) (-(c^2)+(d^2)+2 (-a+c) (x0^2)) (z0^2)))}]

Simplify[(2 x1)/((x1^2)+(((2 x0 x1 y0 ((c-d) (a b-(d^2))+(a-b) (c-d) (x0^2)-(a-b) (c-d) (y0^2)+((a+b) (c+d)-2 (a b+c d)) (z0^2))+2 (c-d) y0 (b d-(d^2)+2 (a-b) (x0^2)+(b-d) ((x0^2)+(y0^2)+(z0^2)))-v2 z0 (-c+(x0^2)+(y0^2)+(z0^2)) (2 x0+x1 (d+(x0^2)+(y0^2)+(z0^2)))))^2)/(4 ((((-c+d) ((b^2)-(d^2)+2 (a-b) (x0^2)) (y0^2)+(b-d) (-(c^2)+(d^2)+2 (-a+c) (x0^2)) (z0^2)))^2))+(((v2 y0 (-b+(x0^2)+(y0^2)+(z0^2)) (2 x0+x1 (d+(x0^2)+(y0^2)+(z0^2)))-2 z0 (-x0 x1 ((b-d) (a c-(d^2))+(b c-2 b d+c d+a (b-2 c+d)) (y0^2)+(a-c) (b-d) ((x0^2)-(z0^2)))+(b-d) ((d^2)-2 a (x0^2)-c (d-(x0^2)+(y0^2)+(z0^2))+d ((x0^2)+(y0^2)+(z0^2))))))^2)/(4 ((((-c+d) ((b^2)-(d^2)+2 (a-b) (x0^2)) (y0^2)+(b-d) (-(c^2)+(d^2)+2 (-a+c) (x0^2)) (z0^2)))^2)))-((8 (C1^2) x1)/(4 (C1^2)(((x1))^2)+(((2 C2+C3 x1))^2)+(((2 C4+C5  x1))^2)))/.{C1->(-c+d) ((b^2)-(d^2)+2 (a-b) (x0^2)) (y0^2)+(b-d) (-(c^2)+(d^2)+2 (-a+c) (x0^2)) (z0^2),C2->-v2 x0 z0 (-c+(x0^2)+(y0^2)+(z0^2))+(c-d) y0 (b d-(d^2)+2 (a-b) (x0^2)+(b-d) ((x0^2)+(y0^2)+(z0^2))),C3->-v2 z0 (-c+(x0^2)+(y0^2)+(z0^2)) (d+(x0^2)+(y0^2)+(z0^2))+2 x0 y0 ((c-d) (a b-(d^2))+(a-b) (c-d) (x0^2)-(a-b) (c-d) (y0^2)+((a+b) (c+d)-2 (a b+c d)) (z0^2)),C4->v2 x0 y0 (-b+(x0^2)+(y0^2)+(z0^2))+(-b+d) z0 ((d^2)-2 a (x0^2)-c (d-(x0^2)+(y0^2)+(z0^2))+d ((x0^2)+(y0^2)+(z0^2))),C5->v2 y0 (-b+(x0^2)+(y0^2)+(z0^2)) (d+(x0^2)+(y0^2)+(z0^2))+2 x0 z0 ((b-d) (a c-(d^2))+(b (c-2 d)+c d+a (b-2 c+d)) (y0^2)+(a-c) (b-d) ((x0^2)-(z0^2)))}]

Therefore

x: x0+(8 (C1^2) x1)/(4 (C1^2)(((x1))^2)+(((2 C2+C3 x1))^2)+(((2 C4+C5  x1))^2))

y:

Simplify[y0+2 y1/(x1^2+y1^2+z1^2)/.{y1->(2(c-d) y0 (b d-d^2+2 (a-b) x0^2+(b -d)((x0^2)+(y0^2)+(z0^2)))+2x0 y0 ((a b-d^2)(c -d )+(a-b)( c-d)(x0^2)-(a-b)( c-d)(y0^2)+((a+b)(c+d)-2 (a b+ c d))(z0^2))x1-v2 z0 (x0^2+y0^2+z0^2-c)((d +(x0^2)+(y0^2)+(z0^2)) x1+2 x0))/(2((y0^2)(d-c )((b^2)-(d^2)+2 (x0^2) (a-b))+(z0^2)(b-d)((d^2) -(c^2)+2(c-a)(x0^2) ))),z1->(-2 z0((b-d) ((d^2)-c d-(2 a-c-d) (x0^2)+(d-c) ((y0^2)+(z0^2)))-((a c-d^2)(b-d)+(b-d)(a-c)(x0^2-z0^2)+( b c-2 b d+d c-2 a c+a b+a d)y0^2)x0  x1)+v2 y0  (-b+(x0^2)+(y0^2)+(z0^2))  (2 x0+(d+(x0^2)+(y0^2)+(z0^2)) x1))/(2 ((-c+d) ((b^2)-(d^2)+2 (a-b) (x0^2)) (y0^2)+(b-d) (-(c^2)+(d^2)+2 (-a+c) (x0^2)) (z0^2)))}]

Simplify[(2 x0 x1 y0 ((c-d) (a b-(d^2))+(a-b) (c-d) (x0^2)-(a-b) (c-d) (y0^2)+((a+b) (c+d)-2 (a b+c d)) (z0^2))+2 (c-d) y0 (b d-(d^2)+2 (a-b) (x0^2)+(b-d) ((x0^2)+(y0^2)+(z0^2)))-v2 z0 (-c+(x0^2)+(y0^2)+(z0^2)) (2 x0+x1 (d+(x0^2)+(y0^2)+(z0^2))))/(((-c+d) ((b^2)-(d^2)+2 (a-b) (x0^2)) (y0^2)+(b-d) (-(c^2)+(d^2)+2 (-a+c) (x0^2)) (z0^2)) ((x1^2)+(((2 x0 x1 y0 ((c-d) (a b-(d^2))+(a-b) (c-d) (x0^2)-(a-b) (c-d) (y0^2)+((a+b) (c+d)-2 (a b+c d)) (z0^2))+2 (c-d) y0 (b d-(d^2)+2 (a-b) (x0^2)+(b-d) ((x0^2)+(y0^2)+(z0^2)))-v2 z0 (-c+(x0^2)+(y0^2)+(z0^2)) (2 x0+x1 (d+(x0^2)+(y0^2)+(z0^2)))))^2)/(4 ((((-c+d) ((b^2)-(d^2)+2 (a-b) (x0^2)) (y0^2)+(b-d) (-(c^2)+(d^2)+2 (-a+c) (x0^2)) (z0^2)))^2))+(((v2 y0 (-b+(x0^2)+(y0^2)+(z0^2)) (2 x0+x1 (d+(x0^2)+(y0^2)+(z0^2)))-2 z0 (-x0 x1 ((b-d) (a c-(d^2))+(b (c-2 d)+c d+a (b-2 c+d)) (y0^2)+(a-c) (b-d) ((x0^2)-(z0^2)))+(b-d) ((d^2)-2 a (x0^2)-c (d-(x0^2)+(y0^2)+(z0^2))+d ((x0^2)+(y0^2)+(z0^2))))))^2)/(4 ((((-c+d) ((b^2)-(d^2)+2 (a-b) (x0^2)) (y0^2)+(b-d) (-(c^2)+(d^2)+2 (-a+c) (x0^2)) (z0^2)))^2))))-(4C1(2 C2+C3 x1)/ (4 (C1^2)(((x1))^2)+(((2 C2+C3 x1))^2)+(((2 C4+C5 x1))^2)))/.{C1->(-c+d) ((b^2)-(d^2)+2 (a-b) (x0^2)) (y0^2)+(b-d) (-(c^2)+(d^2)+2 (-a+c) (x0^2)) (z0^2),C2->-v2 x0 z0 (-c+(x0^2)+(y0^2)+(z0^2))+(c-d) y0 (b d-(d^2)+2 (a-b) (x0^2)+(b-d) ((x0^2)+(y0^2)+(z0^2))),C3->-v2 z0 (-c+(x0^2)+(y0^2)+(z0^2)) (d+(x0^2)+(y0^2)+(z0^2))+2 x0 y0 ((c-d) (a b-(d^2))+(a-b) (c-d) (x0^2)-(a-b) (c-d) (y0^2)+((a+b) (c+d)-2 (a b+c d)) (z0^2)),C4->v2 x0 y0 (-b+(x0^2)+(y0^2)+(z0^2))+(-b+d) z0 ((d^2)-2 a (x0^2)-c (d-(x0^2)+(y0^2)+(z0^2))+d ((x0^2)+(y0^2)+(z0^2))),C5->v2 y0 (-b+(x0^2)+(y0^2)+(z0^2)) (d+(x0^2)+(y0^2)+(z0^2))+2 x0 z0 ((b-d) (a c-(d^2))+(b (c-2 d)+c d+a (b-2 c+d)) (y0^2)+(a-c) (b-d) ((x0^2)-(z0^2)))}]

Therefore
y: y0+4C1(2 C2+C3 x1)/ (4 (C1^2)(((x1))^2)+(((2 C2+C3 x1))^2)+(((2 C4+C5 x1))^2))

z:

Simplify[z0+2 z1/(x1^2+y1^2+z1^2)/.{y1->(2(c-d) y0 (b d-d^2+2 (a-b) x0^2+(b -d)((x0^2)+(y0^2)+(z0^2)))+2x0 y0 ((a b-d^2)(c -d )+(a-b)( c-d)(x0^2)-(a-b)( c-d)(y0^2)+((a+b)(c+d)-2 (a b+ c d))(z0^2))x1-v2 z0 (x0^2+y0^2+z0^2-c)((d +(x0^2)+(y0^2)+(z0^2)) x1+2 x0))/(2((y0^2)(d-c )((b^2)-(d^2)+2 (x0^2) (a-b))+(z0^2)(b-d)((d^2) -(c^2)+2(c-a)(x0^2) ))),z1->(-2 z0((b-d) ((d^2)-c d-(2 a-c-d) (x0^2)+(d-c) ((y0^2)+(z0^2)))-((a c-d^2)(b-d)+(b-d)(a-c)(x0^2-z0^2)+( b c-2 b d+d c-2 a c+a b+a d)y0^2)x0  x1)+v2 y0  (-b+(x0^2)+(y0^2)+(z0^2))  (2 x0+(d+(x0^2)+(y0^2)+(z0^2)) x1))/(2 ((-c+d) ((b^2)-(d^2)+2 (a-b) (x0^2)) (y0^2)+(b-d) (-(c^2)+(d^2)+2 (-a+c) (x0^2)) (z0^2)))}]

Simplify[(v2 y0 (-b+(x0^2)+(y0^2)+(z0^2)) (2 x0+x1 (d+(x0^2)+(y0^2)+(z0^2)))-2 z0 (-x0 x1 ((b-d) (a c-(d^2))+(b (c-2 d)+c d+a (b-2 c+d)) (y0^2)+(a-c) (b-d) ((x0^2)-(z0^2)))+(b-d) ((d^2)-2 a (x0^2)-c (d-(x0^2)+(y0^2)+(z0^2))+d ((x0^2)+(y0^2)+(z0^2)))))/(((-c+d) ((b^2)-(d^2)+2 (a-b) (x0^2)) (y0^2)+(b-d) (-(c^2)+(d^2)+2 (-a+c) (x0^2)) (z0^2)) ((x1^2)+(((2 x0 x1 y0 ((c-d) (a b-(d^2))+(a-b) (c-d) (x0^2)-(a-b) (c-d) (y0^2)+((a+b) (c+d)-2 (a b+c d)) (z0^2))+2 (c-d) y0 (b d-(d^2)+2 (a-b) (x0^2)+(b-d) ((x0^2)+(y0^2)+(z0^2)))-v2 z0 (-c+(x0^2)+(y0^2)+(z0^2)) (2 x0+x1 (d+(x0^2)+(y0^2)+(z0^2)))))^2)/(4 ((((-c+d) ((b^2)-(d^2)+2 (a-b) (x0^2)) (y0^2)+(b-d) (-(c^2)+(d^2)+2 (-a+c) (x0^2)) (z0^2)))^2))+(((v2 y0 (-b+(x0^2)+(y0^2)+(z0^2)) (2 x0+x1 (d+(x0^2)+(y0^2)+(z0^2)))-2 z0 (-x0 x1 ((b-d) (a c-(d^2))+(b (c-2 d)+c d+a (b-2 c+d)) (y0^2)+(a-c) (b-d) ((x0^2)-(z0^2)))+(b-d) ((d^2)-2 a (x0^2)-c (d-(x0^2)+(y0^2)+(z0^2))+d ((x0^2)+(y0^2)+(z0^2))))))^2)/(4 ((((-c+d) ((b^2)-(d^2)+2 (a-b) (x0^2)) (y0^2)+(b-d) (-(c^2)+(d^2)+2 (-a+c) (x0^2)) (z0^2)))^2))))-(4C1(2 C4+C5 x1)/(4 (C1^2)(((x1))^2)+(((2 C2+C3 x1))^2)+(((2 C4+C5 x1))^2)))/.{C1->(-c+d) ((b^2)-(d^2)+2 (a-b) (x0^2)) (y0^2)+(b-d) (-(c^2)+(d^2)+2 (-a+c) (x0^2)) (z0^2),C2->-v2 x0 z0 (-c+(x0^2)+(y0^2)+(z0^2))+(c-d) y0 (b d-(d^2)+2 (a-b) (x0^2)+(b-d) ((x0^2)+(y0^2)+(z0^2))),C3->-v2 z0 (-c+(x0^2)+(y0^2)+(z0^2)) (d+(x0^2)+(y0^2)+(z0^2))+2 x0 y0 ((c-d) (a b-(d^2))+(a-b) (c-d) (x0^2)-(a-b) (c-d) (y0^2)+((a+b) (c+d)-2 (a b+c d)) (z0^2)),C4->v2 x0 y0 (-b+(x0^2)+(y0^2)+(z0^2))+(-b+d) z0 ((d^2)-2 a (x0^2)-c (d-(x0^2)+(y0^2)+(z0^2))+d ((x0^2)+(y0^2)+(z0^2))),C5->v2 y0 (-b+(x0^2)+(y0^2)+(z0^2)) (d+(x0^2)+(y0^2)+(z0^2))+2 x0 z0 ((b-d) (a c-(d^2))+(b (c-2 d)+c d+a (b-2 c+d)) (y0^2)+(a-c) (b-d) ((x0^2)-(z0^2)))}]

Therefore
z: z0+4C1(2 C4+C5 x1)/(4 (C1^2)(((x1))^2)+(((2 C2+C3 x1))^2)+(((2 C4+C5 x1))^2))

Hence by setting
r->2 C1, U1->0,U2->2 C2, U3->2 C4,V1->2 C1, V2->C3,V3->C5, we can apply Lemma.

Thus  t is

Simplify[-((-U3 V2+U2 V3)/(U3 V1-U1 V3))/.{ U1->0,U2->2 C2, U3->2 C4,V1->2 C1, V2->C3,V3->C5}]

Simplify[(C3 C4-C2 C5)-2 C1(2(b-c)(a-d)x0 y0 z0+v2(-b y0^2-c z0^2+(x0^2+y0^2+z0^2)(y0^2+z0^2)))/.{C1->(-c+d) ((b^2)-(d^2)+2 (a-b) (x0^2)) (y0^2)+(b-d) (-(c^2)+(d^2)+2 (-a+c) (x0^2)) (z0^2),C2->-v2 x0 z0 (-c+(x0^2)+(y0^2)+(z0^2))+(c-d) y0 (b d-(d^2)+2 (a-b) (x0^2)+(b-d) ((x0^2)+(y0^2)+(z0^2))),C3->-v2 z0 (-c+(x0^2)+(y0^2)+(z0^2)) (d+(x0^2)+(y0^2)+(z0^2))+2 x0 y0 ((c-d) (a b-(d^2))+(a-b) (c-d) (x0^2)-(a-b) (c-d) (y0^2)+((a+b) (c+d)-2 (a b+c d)) (z0^2)),C4->v2 x0 y0 (-b+(x0^2)+(y0^2)+(z0^2))+(-b+d) z0 ((d^2)-2 a (x0^2)-c (d-(x0^2)+(y0^2)+(z0^2))+d ((x0^2)+(y0^2)+(z0^2))),C5->v2 y0 (-b+(x0^2)+(y0^2)+(z0^2)) (d+(x0^2)+(y0^2)+(z0^2))+2 x0 z0 ((b-d) (a c-(d^2))+(b (c-2 d)+c d+a (b-2 c+d)) (y0^2)+(a-c) (b-d) ((x0^2)-(z0^2)))}/.d->(Sqrt[2 a x0^2+2 b y0^2+2 c z0^2-(x0^2+y0^2+z0^2)^2])]

Therefore

t:((C3 C4-C2 C5)/(2 C1 C4))=(2(b-c)(a-d)x0 y0 z0+v2(-b y0^2-c z0^2+(x0^2+y0^2+z0^2)(y0^2+z0^2)))/C4

s is

Simplify[-((U2 V1-U1 V2)/(-U3 V1+U1 V3))/.{ U1->0,U2->2 C2, U3->2 C4,V1->2 C1, V2->C3,V3->C5}]

The partial derivatives of z by x,y  are  zx->x0(a-x0^2-y0^2-z0^2)/(z0(x0^2+y0^2+z0^2-c)),zy->y0(b-x0^2-y0^2-z0^2)/(z0(x0^2+y0^2+z0^2-c)).

Then T:=(t+s zx)/(1-s zy)

Solve[(x0^2+y0^2+z0^2)^2-2 a x0^2-2 b y0^2-2 c z0^2+d^2==0,z0]

When z0->(Sqrt[c-(x0^2)-(y0^2)+(Sqrt[(c^2)-(d^2)+2( a-c) (x0^2)+2 (b-c) (y0^2)])]), we show that  T=P/Q with

Q:=-(b-d) (((c-d))^2) (2 c+d)(c+d)+(b-d) (-c+d) (6 a c-7 (c^2)+4 (a -c )d+(d^2))(x0^2) -(b-c) (b-d) (c-d) (3 c+d) (y0^2)-2 (a-c) (b-d) (2 a-3 c+d) (x0^4)+ 2 (b-c) (-2 a b+a c+2 b c+(a-3 c) d+(d^2))(x0^2) (y0^2)-(Sqrt[(c^2)-(d^2)+2( a-c) (x0^2)+2 (b-c) (y0^2)])((b-d) (c-d) (2 c-d) (c+d)+(b-d) (4 a c-5 (c^2)-2 (a -c) d+(d^2))(x0^2)+(b-c) (b-d) (c-d) (y0^2)-2 (a-c) (b-d) (x0^4)- 2 (b-c) (a-d)(x0^2) (y0^2))

P:=x0 y0(-((c^2)-(d^2)) (3 a b-2 a c-2 b c-(a +b -4 c )d-(d^2))+2 (a-c) (-2 a b+a c+2 b c+(a -3 c) d+(d^2))x0^2+2 (b-c) (-2 a b+2 a c+b c+(b -3 c) d+(d^2))y0^2+(Sqrt[(c^2)-(d^2)+2( a-c) (x0^2)+2 (b-c) (y0^2)])(-3 a b c+2 a (c^2)+2 b (c^2)+(a b +a c +b c-4 (c^2) )d-(a +b -c )(d^2)+(d^3)+2 (a-c) (b-d) (x0^2)+2 (b-c) (a-d) (y0^2)))-v2 (Sqrt[c-(x0^2)-(y0^2)+(Sqrt[(c^2)-(d^2)+2( a-c) (x0^2)+2 (b-c) (y0^2)])])(c((c^2)-(d^2)+2 (a-c) (x0^2)+2 (b-c) (y0^2))+(x0^2+y0^2+z0^2-c)((c^2)-(d^2)+ (a-c) (x0^2)+ (b-c) (y0^2)))

Simplify[(t+s zx)/(1-s zy)-P/Q/.{t->(2(b-c)(a-d)x0 y0 z0+v2(-b y0^2-c z0^2+(x0^2+y0^2+z0^2)(y0^2+z0^2)))/C4, s->C2/C4, zx->x0(a-x0^2-y0^2-z0^2)/(z0(x0^2+y0^2+z0^2-c)),zy->y0(b-x0^2-y0^2-z0^2)/(z0(x0^2+y0^2+z0^2-c))}/.{C2->-v2 x0 z0 (-c+(x0^2)+(y0^2)+(z0^2))+(c-d) y0 (b d-(d^2)+2 (a-b) (x0^2)+(b-d) ((x0^2)+(y0^2)+(z0^2))),C4->v2 x0 y0 (-b+(x0^2)+(y0^2)+(z0^2))+(-b+d) z0 ((d^2)-2 a (x0^2)-c (d-(x0^2)+(y0^2)+(z0^2))+d ((x0^2)+(y0^2)+(z0^2)))}/.{z0->(Sqrt[c-(x0^2)-(y0^2)+(Sqrt[(c^2)-(d^2)+2( a-c) (x0^2)+2 (b-c) (y0^2)])]),v2->(Sqrt[2(b-d) (c-d)  (d-a)])}]

The center of the circle is

x:

Simplify[x0+(r (-U2 V1 V2+U1 (V2^2)-U3 V1 V3+U1 (V3^2)))/((U2^2) (V1^2)+(U3^2) (V1^2)-2 U1 U2 V1 V2+(U1^2) (V2^2)+(U3^2) (V2^2)-2 U1 U3 V1 V3-2 U2 U3 V2 V3+(U1^2) (V3^2)+(U2^2) (V3^2))/.{r->2 C1, U1->0,U2->2 C2, U3->2 C4,V1->2 C1, V2->C3,V3->C5}]

y:

Simplify[y0+(r (U2 (V1^2)-U1 V1 V2-U3 V2 V3+U2 (V3^2)))/((U2^2) (V1^2)+(U3^2) (V1^2)-2 U1 U2 V1 V2+(U1^2) (V2^2)+(U3^2) (V2^2)-2 U1 U3 V1 V3-2 U2 U3 V2 V3+(U1^2) (V3^2)+(U2^2) (V3^2))/.{r->2 C1, U1->0,U2->2 C2, U3->2 C4,V1->2 C1, V2->C3,V3->C5}]

Simplify[z0+(r (U3 (V1^2)+U3 (V2^2)-U1 V1 V3-U2 V2 V3))/((U2^2) (V1^2)+(U3^2) (V1^2)-2 U1 U2 V1 V2+(U1^2) (V2^2)+(U3^2) (V2^2)-2 U1 U3 V1 V3-2 U2 U3 V2 V3+(U1^2) (V3^2)+(U2^2) (V3^2))/.{r->2 C1, U1->0,U2->2 C2, U3->2 C4,V1->2 C1, V2->C3,V3->C5}]

(The radius)^2 is

Simplify[((r^2) ((V1^2)+(V2^2)+(V3^2)))/((U3^2) ((V1^2)+(V2^2))-2 U1 U3 V1 V3-2 U2 V2 (U1 V1+U3 V3)+(U2^2) ((V1^2)+(V3^2))+(U1^2) ((V2^2)+(V3^2)))/.{r->2 C1, U1->0,U2->2 C2, U3->2 C4,V1->2 C1, V2->C3,V3->C5}]