Since conformal space is 2 dimensions higher than the space we are working in, that means to model 3D Euclidean space, we need 5D Conformal space.

## Example

So if we are initially at point (x=3, y=4, z=5) this will be represented by the multivector:

P1 = -n_{0}+ 50 n_{∞} + 6 n_{1} + 8 n_{2} + 10 n_{3}

A pure displacement of (x=4, y=2, z=6) will be represented by the multivector:

1 + 2 _{} n_{∞1} + 1 _{} n_{∞2}+ 3 _{} n_{∞3}

So to combine these, to give the resulting position, we use P2 = m * P1 * m' to give:

p2 = (1 + 2 _{} n_{∞1} + 1 _{} n_{∞2}+ 3 _{} n_{∞3})*(-n_{0}+ 50 n_{∞} + 6 n_{1} + 8 n_{2} + 10 n_{3})*(1 - 2 _{} n_{∞1} - 1 _{} n_{∞2}- 3 _{} n_{∞3})

multiplying out the first two terms using the above multiplication table we get (using the rules explained on this page):

P2 = (-n_{0}+ 50 n_{∞} + 3 n_{1} + 4 n_{2} + 5 n_{3}

-4n_{∞1}n_{0}+ 200 n_{∞1}n_{∞} + 12 n_{∞1}n_{1} + 16 n_{∞1}n_{2} + 20 n_{∞1}n_{3}

-2n_{∞2}n_{0}+ 100 n_{∞2}n_{∞} + 6 n_{∞2}n_{1} + 8 n_{∞2}n_{2} + 10 n_{∞2}n_{3}

-6n_{∞3}n_{0}+ 300 n_{∞3}n_{∞} + 18 n_{∞3}n_{1} + 24 n_{∞3}n_{2} + 30 n_{∞3}n_{3})*(1 - 4 _{} n_{∞1} - 2 _{} n_{∞2}- 6 _{} n_{∞3})

P2 = (-n_{0}+ 50 n_{∞} + 3 n_{1} + 4 n_{2} + 5 n_{3}

+4n_{∞01}+ 12 n_{∞}+ 16 n_{∞1}_{2} - 20 n_{∞31}

+2n_{∞02}- 6 n_{∞12}+ 8 n_{∞}+ 10 n_{∞2}_{3}

+6n_{∞03}+ 18 n_{∞3}_{1} - 24 n_{∞23}+ 30 n_{∞})*(1 - 4 _{} n_{∞1} - 2 _{} n_{∞2}- 6 _{} n_{∞3})

P2 = (-n_{0}+ (50+12+8+30) n_{∞} + 3 n_{1} + 4 n_{2} + 5 n_{3}

+4(n_{1}-n_{0∞1}) + (16-6) n_{∞1}_{2} +(18- 20) n_{∞31} +2(n_{2}-n_{0∞2}) + (10-24) n_{∞2}_{3}

+6(n_{3}-n_{0∞3}))*(1 - 4 _{} n_{∞1} - 2 _{} n_{∞2}- 6 _{} n_{∞3})

P2 = (-n_{0}+100n_{∞}+7n_{1}+6n_{2}+11 n_{3}-4 n_{0∞1}-2 n_{0∞2}-6 n_{0∞3}+10 n_{∞1}_{2}-2 n_{∞31}-14 n_{∞2}_{3}

)*(1 - 4 _{} n_{∞1} - 2 _{} n_{∞2}- 6 _{} n_{∞3})

P2 = -n_{0}+100n_{∞}+7n_{1}+6n_{2}+11 n_{3}-4 n_{0∞1}-2 n_{0∞2}-6 n_{0∞3}+10 n_{∞12}-2 n_{∞31}-14 n_{∞2}_{3
}+4n_{0}n_{∞1}-400n_{∞}n_{∞1}-28n_{1}n_{∞1}-24n_{2}n_{∞1}-44 n_{3}n_{∞1}+16 n_{0∞1}n_{∞1}+8 n_{0∞2}n_{∞1}+24 n_{0∞3}n_{∞1}-40 n_{∞1}_{2}n_{∞1}+8 n_{∞31}n_{∞1}+56 n_{∞2}_{3}n_{∞1}

+2n_{0}n_{∞2}-200n_{∞}n_{∞2}-14n_{1}n_{∞2}-12n_{2}n_{∞2}-22 n_{3}n_{∞2}+8 n_{0∞1}n_{∞2}+4 n_{0∞2}n_{∞2}+12 n_{0∞3}n_{∞2}-20 n_{∞1}_{2}n_{∞2}+4 n_{∞31}n_{∞2}+28 n_{∞2}_{3}n_{∞2}

+6n_{0}n_{∞3}-600n_{∞}n_{∞3}-42n_{1}n_{∞3}-36n_{2}n_{∞3}-66 n_{3}n_{∞3}+24 n_{0∞1}n_{∞3}+12 n_{0∞2}n_{∞3}+36 n_{0∞3}n_{∞3}-60 n_{∞1}_{2}n_{∞3}+12 n_{∞31}n_{∞3}+72 n_{∞2}_{3}n_{∞3}

P2 = -n_{0}+(100+66+28+12)n_{∞}+7n_{1}+6n_{2}+11n_{3}+(-4+4) n_{0∞1}+(2-2) n_{0∞2}+(6-6)n_{0∞3}+(10-24+14) n_{∞1}_{2}+(44-2-42) n_{∞31}(36-14-22) n_{∞2}_{3
}

P2 = -n_{0}+206 n_{∞}+7 n_{1}+6 n_{2}+11 n_{3}

### Pure Rotation (no displacement)

If the displacement is zero then n_{1},n_{2},n_{3} = 0 and the rotation is represented by the multivector: cos(θ/2) + a_{x}sin(θ/2) n_{23} + a_{y}sin(θ/2) n_{31} + a_{z}sin(θ/2) n_{12}

#### Example

Applying a rotation of point (3,4,5) by 180° around the x axis is given by:

P2 = (n_{23})*(-n_{0}+ 50 n_{∞} + 6 n_{1} + 8 n_{2} + 10 n_{3})*(-n_{23})

P2 = (-n_{23}n_{0}+ 50 n_{23}n_{∞} + 6 n_{23}n_{1} + 8 n_{23}n_{2} + 10 n_{23}n_{3})*(-n_{23})

P2 = (-n_{023}+ 50 n_{∞23} + 6 n_{123} - 8 n_{3}+ 10 n_{2})*(-n_{23})

P2 = n_{023}n_{23}- 50 n_{∞23}n_{23} - 6 n_{123}n_{23} + 8 n_{3}n_{23}- 10 n_{2}n_{23}

P2 = -n_{0} + 50 n_{∞}+ 6 n_{1}- 8 n_{2}- 10 n_{3}

### Combined Displacement and Rotation (displace then rotation)

#### Example

Starting from the previous position: (-n_{0}+ 50 n_{∞} + 6 n_{1} + 8 n_{2} + 10 n_{3})

and both displace by (x=4, y=2, z=6) and applying a rotation of 180° around the x axis represented by: (n_{23})

Therefore:

m = (n_{23}) (1 + 4 _{} n_{∞1} + 2 _{} n_{∞2}+ 6 _{} n_{∞3})

m = 1 n_{23} + 4 _{} n_{23}n_{∞1} + 2 _{} n_{23}n_{∞2}+ 6 _{} n_{23}n_{∞3}

m = 1 n_{23} + 4 _{}_{∞123} - 2 _{}n_{∞3}+ 6 _{}n_{∞2}

So applying the transform gives:

P2 = (n_{23} + 4 _{}_{∞123} - 2 _{}n_{∞3}+ 6 _{}n_{∞2})*(-n_{0}+ 50 n_{∞} + 3 n_{1} + 4 n_{2} + 5 n_{3})*(n_{23} + 4 _{}_{∞123} - 2 _{}n_{∞3}+ 6 _{}n_{∞2})

P2 = (-n_{0}+ 206 n_{∞} + 7 n_{1} -6 n_{2} -11 n_{3})