{ "blyg": "0.4", "id": "3va8ms0b2jzrprh1zp7a95y46a", "kind": "thread", "origin": "https://notionparallax.co.uk/", "page": "https://notionparallax.co.uk/2008/gc-strategies-ensuring-planarity-with-sheared-cones", "author": { "name": "Ben Doherty", "url": "https://notionparallax.co.uk/" }, "created": "2008-04-17T17:28:13Z", "updated": "2008-04-17T17:28:13Z", "version": 1, "content_md": "# gc strategies - ensuring planarity with sheared cones\n\n\n

here's another strategies tutorial, there is a pdf and gct that's more beautifully formatted over at GCuser.com in the gallery.

\n

This is about making flat quad panels using sheared cones.

\n

For those of us in Melbourne, there is a particularly horrific example of this in Melbourne central. if you look carefully, you can see the module size split halfway up when the panels get too small.

\n

\n

\n

The default solution to panelling anything is to do it in triangles. Any surface can be tiled in triangles. Any three points in space define a triangle, and also a plane.
\nQuadrilateral panels are a bit trickier as there are only a certain number of ways of making a surface be predictably tillable using four sided shapes.
\nThe way outlined here is the sheared cones method.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
Given that all triangles are planar, and
\nsubdivision of that triangle is guaranteed to be
\nplanar too.
\nIf the base of a cone is chopped so that it has flat
\nsides then the sides of the cone become
\ntriangles.
If we then slice the top off the cone, parallel to
\nthe base, then the remaining sections of panel
\nare still going to be flat.
As counter intuitive as it initially feels, if we move
\nthe top point of the cone around then the panels
\nremain flat as the triangles are still intact. So
\nbasically, shearing the cone has no effect on the
\nplanarity, hence the name “sheared cone”
\nmethod.
So if we bump up the number of sides on the
\ncone, we begin to see that we are beginning to
\nget a more realistic panellisation of the circle.
The top set of points can then become the
\nbottom set of points for a new set of panels, and
\nso on.
So in summary, if two circles are parallel, and
\nthey have the same number of points, equally
\nspaced on their circumference, then the panels
\nbetween them will be planar, regardless of the
\nshear or scale of the circles with respect to each
\nother.
\n

\"Creative
\nSheared cones tutorial by Ben Doherty is licensed under a Creative Commons Attribution-Share Alike 2.0 UK: England & Wales License.

\n", "content_html": "

gc strategies - ensuring planarity with sheared cones

\n

here’s another strategies tutorial, there is a pdf and gct that’s more beautifully formatted over at GCuser.com in the gallery.

\n

This is about making flat quad panels using sheared cones.

\n

For those of us in Melbourne, there is a particularly horrific example of this in Melbourne central. if you look carefully, you can see the module size split halfway up when the panels get too small.

\n

\n

\n

The default solution to panelling anything is to do it in triangles. Any surface can be tiled in triangles. Any three points in space define a triangle, and also a plane.
\nQuadrilateral panels are a bit trickier as there are only a certain number of ways of making a surface be predictably tillable using four sided shapes.
\nThe way outlined here is the sheared cones method.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
Given that all triangles are planar, and
\nsubdivision of that triangle is guaranteed to be
\nplanar too.
\nIf the base of a cone is chopped so that it has flat
\nsides then the sides of the cone become
\ntriangles.
If we then slice the top off the cone, parallel to
\nthe base, then the remaining sections of panel
\nare still going to be flat.
As counter intuitive as it initially feels, if we move
\nthe top point of the cone around then the panels
\nremain flat as the triangles are still intact. So
\nbasically, shearing the cone has no effect on the
\nplanarity, hence the name “sheared cone”
\nmethod.
So if we bump up the number of sides on the
\ncone, we begin to see that we are beginning to
\nget a more realistic panellisation of the circle.
The top set of points can then become the
\nbottom set of points for a new set of panels, and
\nso on.
So in summary, if two circles are parallel, and
\nthey have the same number of points, equally
\nspaced on their circumference, then the panels
\nbetween them will be planar, regardless of the
\nshear or scale of the circles with respect to each
\nother.
\n

\"Creative
\nSheared cones tutorial by Ben Doherty is licensed under a Creative Commons Attribution-Share Alike 2.0 UK: England & Wales License.

\n", "content_hash": "sha256:ffdd257aa51ac689dfe39bb0c1904cfb86ff6d04539d6b42b4ef544f19c15faa", "media": [], "changelog": [ { "version": 1, "at": "2008-04-17T17:28:13Z", "note": null } ], "transclusions": [] }