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Re: Fiting squares under a curve.

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Malcolm McLean <malcolm.arthur.mclean@gmail.com> writes:

> On Wednesday, 15 September 2021 at 15:17:39 UTC+1, Ben Bacarisse wrote:
>> Malcolm McLean <malcolm.ar...@gmail.com> writes: 
>> 
>> > Maybe there's enough theoretical content here to justify a post to 
>> > comp.theory, though it's largely a practical problem. 
>> > 
>> > I have an image made up of a frieze of square cells. I can add cells 
>> > at will to the left or the rich (within reason). I want to fit it to a 
>> > wall with a curved top, retaining the aspect ratio as much as 
>> > possible. So small cells in low parts of the wall, large cells in high 
>> > parts. 
>> > 
>> > The flat top case is easy. The cell width is given by the height of 
>> > the wall to make a square. It is then rounded to make a whole number 
>> > of cells. As long as the wall is longer than it is high, the worst 
>> > case is a 150% scaling in the x direction. Normally the wall is much 
>> > longer than high, and distortion is negligible. 
>> > 
>> > However I want to allow a curved top. So the cells are no longer of 
>> > the same height, in fact they are quads with two right angles at the 
>> > bottom and a sloping top. The stright line function of the cell tops 
>> > approximates the wall top curve. 
>> > 
>> > if we consider the solution to be a set of points in x along the wall 
>> > representing cells, and the midpints the boundaries of cells, then we 
>> > need to minimise the function 
>> > 
>> > sum over each cell : integral of wall function from cell x low to cell 
>> > x high minus (cell x high - cell x low) squared. 
>> > 
>> > Is there a fast way of doing this?
>> It might be useful to know why I didn't comment before in 
>> comp.programming: I didn't understand what the constraints are. 
>> 
>> You can add cells to some already given strip of cells but what is "with 
>> reason"? Can you add cells from some restricted set, or can you choose 
>> everything about the cells you add? Can you choose anything about the 
>> initial frieze? 
>>
> The user specifies the contents of the cell, which can be any image. In fact
> it's a vector image. I then construct a frieze from the cell by adding as many
> cells as are needed to make up the length of the wall (in the flat top case).
> Often the images will have geometry which joins. So two adjoining cells must 
> have a common edge of the same height. I was planning to make the cells
> into quads with two ritch angles at the bottom and a slanted top to accommodate
> this requirement.
> I can add as many cells as I want to the frieze, until I overload the capacity of the
> various systems to handle the geometry. Which means about a hundred cells 
> would be a limit. There's also no point in having a very tiny cell. The end user can't 
> see sub-pixel geometry, it's waste of resources calculating it.
>> 
>> You can scale something. It is the function to be fitted or the frieze 
>> of cells? Scaling implies an origin. What is that origin and/or can 
>> the scaling be accompanied by a translation making the origin 
>> unimportant? 
>> 
> I can't scale the wall to the end user. But of course I can scale intermediates,
> for instance to trivially transform the rectangular cell problem into the square 
> cell problem by strething or contracting the wall. Each cell is small square
> image. I was thinking of simply scaling the whole cell by a factor, but of course
> this isn't strictly correct - internally different regions of the cell have different
> scales because the wall is not flat over the cell. I'm hoping this won't matter
> too much visually.
>   
>> Do you have any extra information about the function? For example, is 
>> it everywhere continuous and maybe even smooth? 
>> 
> The function was chosen to be as close as possible to an existing industry-
> standard "variable width profile" which we can manipulate, but the equations 
> are not public. The user enters "width points" and drags them out and slides 
> them, and the profile interpolates. I ended up using quintic interpolation,
> with special cases for when two width points  are at identical x positions.
> So you can create steps and straight line sections, but normally the
> function is continuous and smooth for its entire length. (I can also
> restrict the function as a last resort, I coded it and have control of it, but
> then it would no longer be as similar to the function in the industry-
> standard tool).

I'm sorry but I can't work out what the optimisation problem is from
this description.  Also, the problem is described in domain-specific
jargon which makes it hard for outsiders.

-- 
Ben.

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Fiting squares under a curve. Malcolm McLean <malcolm.arthur.mclean@gmail.com> - 2021-09-15 06:59 -0700
  Re: Fiting squares under a curve. Ben Bacarisse <ben.usenet@bsb.me.uk> - 2021-09-15 15:17 +0100
    Re: Fiting squares under a curve. Malcolm McLean <malcolm.arthur.mclean@gmail.com> - 2021-09-15 08:06 -0700
      Re: Fiting squares under a curve. Ben Bacarisse <ben.usenet@bsb.me.uk> - 2021-09-15 22:33 +0100
        Re: Fiting squares under a curve. Malcolm McLean <malcolm.arthur.mclean@gmail.com> - 2021-09-15 15:01 -0700
          Re: Fiting squares under a curve. Ben Bacarisse <ben.usenet@bsb.me.uk> - 2021-09-16 00:33 +0100
            Re: Fiting squares under a curve. Malcolm McLean <malcolm.arthur.mclean@gmail.com> - 2021-09-17 03:25 -0700
              Re: Fiting squares under a curve. Ben Bacarisse <ben.usenet@bsb.me.uk> - 2021-09-17 14:22 +0100
                Re: Fiting squares under a curve. Andy Walker <anw@cuboid.co.uk> - 2021-09-17 16:25 +0100
                Re: Fiting squares under a curve. Ben Bacarisse <ben.usenet@bsb.me.uk> - 2021-09-17 20:59 +0100
                Re: Fiting squares under a curve. Malcolm McLean <malcolm.arthur.mclean@gmail.com> - 2021-09-23 02:14 -0700

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