A method for computer generated Celtic knot patterns. This prototype tool was built using Houdini software. The implemented algorithm is based on a paper titled
Computer Generated Celtic Design, by Matthew Kaplan and Elaine Cohen.
As always the project files are available for members of my Pateron page.
The interpolation step in the algorithm involves
cubic Hermite interpolation, a specific method chosen due to its requirement for only specifying endpoint vectors and their tangents.
Consider p0 as the starting point in the image below. According to the algorithm, the endpoint is p1, located at the midpoint of the next clockwise edge. It's important to note that if we initiate the direction along the right outward vector (m0), the tangent at our endpoint should align with the left inward vector (m1). Once the variables p0, p1, m0, and m1 are established, the interpolation can be done using the formula below.
$\overrightarrow{p}(t) = (2t^3 - 3t^2 + 1)\overrightarrow{p0} + (t^3 - 2t^2 + t)\overrightarrow{m0}$
$ + (-2t^3 +3t^2)\overrightarrow{p1}+ (t^3 - t^2)\overrightarrow{m1}$
where $t \in [0,1]$
$\overrightarrow{p}(t) = (2t^3 - 3t^2 + 1)\overrightarrow{p0} + (t^3 - 2t^2 + t)\overrightarrow{m0} + (-2t^3 +3t^2)\overrightarrow{p1}+ (t^3 - t^2)\overrightarrow{m1}$
where $t \in [0,1]$
