# RE: things often done in geometrical applications

From: Dailey, David P. <david.dailey@sru.edu>
Date: Fri, 26 Jul 2013 08:59:40 -0400
To: Doug Schepers <schepers@w3.org>, David Dailey <ddailey@zoominternet.net>

```A couple of other things that would be nice
1. Given an arbitrary convex quadrilateral, return the unique ellipse tangent to the four sides:
http://chrisjones.id.au/Ellipses/ellipse.html -- useful for perspective drawing.

2. Given two circles return the four lines tangent to both (in the typical situation of disjoint circles)

http://jwilson.coe.uga.edu/emt669/Student.Folders/Kertscher.Jeff/Essay.3/Tangents.html
and
http://cs.sru.edu/~ddailey/svg/circlesX.svg
(Jim Wilson at UGA tells me that this technique was undoubtedly known by Euclid and students)

3.  Given a line arrangement ( https://en.wikipedia.org/wiki/Arrangement_of_lines ) return the collection of polygons into which the lines have divided the viewport.

4. Given three (nonoverlapping) circles, return the eight Apollonian circles tangent to all three
http://en.wikipedia.org/wiki/Problem_of_Apollonius

That these problems and their solutions have endured for millennia is plenty of justification for their utility to me.

cheers
David
________________________________________
From: Doug Schepers [schepers@w3.org]
Sent: Monday, July 01, 2013 1:25 AM
To: David Dailey
Cc: public-geometryapi@w3.org
Subject: Re: things often done in geometrical applications

Hi, David-

On 6/28/13 4:06 PM, David Dailey wrote:
>
> Thanks for getting this going. Coincidentally, as it’s creation was
> being discussed, I in the midst of a project that could have used the
> fruits of this labor (assuming they might exist in a couple of years).

I've seen so many requests for something like this over the years that
it was bound to overlap several people's projects that need it. :)

Thanks for this list to help us get started. I have a few comments on
specific items, inline...

> Here’s just a quick list of things that come to mind that one might
> often wish to do
>
> Intersecting things, as per Kevin Lindsay’s work (wasn’t some of this
>
>                  Lines, circles, ellipses, bezier (c and q) all with one
> another

I've added this to the use cases and requirements page [1]. Note that
Kevin has pointed out some flaws and limitations in his library, and in
using it, I've found that you have to know the segment/line type before
performing an intersection test, which limits its usefulness.

This makes it a good candidate for addressing in a browser-native API,
in my opinion.

> Points closest to one another  on various pairs of curves (simplest
> being two non parallel lines or a line and circle)
>
> Lines tangent to a given curve
>
>                  Closest to a point
>
>                  At a point
>
>                  Passing through a poin
>
> Lines tangent to each of two
>
>                  Circles (see
> http://jwilson.coe.uga.edu/emt669/Student.Folders/Kertscher.Jeff/Essay.3/Tangents.html
> )
>
>                  Ellipses

I could see some of this being useful in a drawing tool.

> Tweening – interpolating curves that are intermediate in shape between
> two given curves (see the code that comes with <replicate>

Kevin had a library (or script) for this as well, but I agree it could
spawn some good requirements.

> Extrapolation – given curves A and B produce C so that B is 1/nth of the
> way from A to C.
>
> Given a set of n lines that divide the plane into (n^2 + n +2)/2 regions
> – convert those regions to convex polygons returning a collection of
> polygons
>
> Given a smooth path, convert it to a polygon having n edges for given
> n>2 most closely resembling the path.
>
> Given a polygon (convex or concave) or more generally,
>
>                  Form a triangulation of its interior
>
>                  Form a slight deformation of it (for glyphs we might
> consider arbitrary  svg <path>s rather than just polygons

You've lost me here. What sort of use cases would these address? Even
just some illustrative examples would help.

>                  Produce a smoothed version (see
> http://srufaculty.sru.edu/david.dailey/svg/Draw018.html -- draw a
> rectangle, select it (by clicking on it) and then choose Smooth from the

Smoothing curves/shapes is really useful for drawing apps, especially
those that have a distributed shared canvas, since it results in simpler
paths to exchange.

> Given n points
>
>                  Find a polygon passing through them
>
>                  Produce a Voronoi diagram
>
> Produce random
>
>                  N-gon
>
>                  Smooth Bezier curves that pass through N control points
> and preserve continuous first derivatives
>
>                  Tessellations (polygonal, or rectangular – think HTML
> tables with colspan and rowspan)

These seem more like things you'd do in script, maybe using a geometry
API for parts, rather than operations of a geometry API itself. But
maybe I'm missing something.

> That such things are useful (hence implying the existence of use cases)
> seems obvious through a simple glance at the last 2500 years of progress
> in algebra, mensuration and geometry.

We'll have a much stronger case to bring to the browser implementers if
we reinforce our requirements with explicit use cases. It will also help
us focus on the highest-priority items.

> BTW, Inkscape has some of this stuff already, as does <replicate>,
> neither of which I saw in the script library listed in the wiki.

Like it says on the wiki, that page was a rough first draft, and anyone
in the Community Group can edit it.

(That said, Inkscape isn't a script library... but all those links were
just for inspiration anyway.)

[1] http://www.w3.org/community/geometryapi/wiki/Use_Cases_and_Requirements

Regards-
-Doug
```
Received on Friday, 26 July 2013 13:00:10 UTC

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