Reciprocal System Research Society

Bruce,

Some parameric curves involve a ratio of two magintudes (similar to space and time) coupled via simple functions.

Look at the brief description of 3D NURBS below, which seem to couple 3D magnitude to 1D magnitude in their implementation.

It seems that with some minor tweaking their definition can be expanded to 3D/3D magnitudes and used to model RST. (we'd need three different W(t) functions, to be symmetrical)

Don't forget to read the last 5 words of the quote below.

Regards,

Horace

Google wrote:

A rational cubic curve segment in 3D can be constructed as follows

x(t) = X(t)/W(t)

y(t) = Y(t)/W(t)

z(t) = Z(t)/W(t)

where each of X(t), Y(t), Z(t), and W(t) are cubic polynomial curves. Defining curves as rational polynomials in this manner allows for simple exact represenations of conic sections such as circles, as well as curves which are invariant under perspective projection.

Horace wrote:

Google wrote:

A rational cubic curve segment in 3D can be constructed as follows

x(t) = X(t)/W(t)

y(t) = Y(t)/W(t)

z(t) = Z(t)/W(t)

where each of X(t), Y(t), Z(t), and W(t) are cubic polynomial curves. Defining curves as rational polynomials in this manner allows for simple exact representations of conic sections such as circles, as well as curves which are invariant under perspective projection.

The homogeneous coordinate representation would be: | X(t) Y(t) Z(t) W(t) |

It is only invariant under metric and Euclidean strata, where perspective transform takes place. (Affine would have three, independent quantities--you would need 3 separate W functions).

But dang, that's rather a nifty solution to modeling that locus of points (mesh surface) that results from the discrete unit postulate when dealing with an aggregate. One polynomial is a lot easier to deal with than a gazillion points.

The Universe is looking more and more like a computer model every day... maybe we are just characters in a holo-novel. Now if I could just figure out how to call for the "arch"...

Thanks!

It should be elementary my dear Doctor, elementary...

Dr Moriarty wrote:

Now if I could just figure out how to call for the "arch"...

I wonder what would be the appearance of a plot depicting all the NURB curves satisfying a constant cross-ratio constraint ?