Interesting contrast to this is that most modern vector 2D graphics API use PostScript/PDF drawing model that cannot represent exact circles and elipses and these are instead interpolated by a bunch of bezier curve segments.
Bezier curves (the regular kind) cannot represent circles and ellipses, this is true. However, rational Bezier curves can [1]. I don't know if PostScript has a rational Bezier primitive, but some drawing libraries do. A prominent example of a library that does is Skia, the rendering engine behind Chrome and Android.
I don't think that's true. Drawing Bezier curves is not any easier than drawing ellipses, and in fact I'm pretty sure it's even more difficult. If someone wants to approximate an ellipse the easiest solution is to just draw a bunch of short segments.
Can you provide examples? I have not seen an API not to support circles and arcs. SVG is AFAIK the most common API, you have there circles, arcs, ellipsis, quadratic and cubic beziers etc.
This is how SVG and most other drawing programs are implemented under the hood.
Down at the lowest level the engine can only really do straight lines. You may ask for a circle or a bezier, but the engine only approximates this with enough fidelity that you wouldn't notice the flaws. Beziers end up being approximated as a bunch of tiny line segments. Circles and ellipses can be approximated pretty easily as a handful of beziers.
Beziers are particularly useful because they have some handy properties. You don't need to do any trig to calculate them (like you do for arcs). They're extremely simple to compute, and any bezier curve can be split into two smaller bezier curves. That last fact makes it easier to have varying fidelity along the curve.
Source: was on the original SVG Working Group team @ Adobe.
FWIW, here's the bezier algorithm in Python:
def bezier(t, *handles):
"""Computes a single point of a bezier of any order where 0 <= t <= 1."""
if len(handles) == 1:
return handles[0]
else:
pairs = zip(handles[:-1], handles[1:])
return bezier(t, *[_bezier_interpolate(t, a, b) for a, b in pairs])
def _bezier_interpolate(t, a, b):
return [a * (1-t) + b * t for a, b in zip(a, b)]
assert bezier(0.25, [1, 1, 1], [2, 2, 2], [3, 4, 5]) == [1.5, 1.5625, 1.625]
You can pass it any order of bezier (cubic, quadratic, etc.) for any number of dimensions, and how for along the interpolation you are. It will give you a single point. Connect a bunch together to get your curve. You can actually stop when `len(handles) == 2` and you get line segments instead of points, but it's slightly less accurate.
You can see the math for computing a point (or segment) is trivial (addition and multiplication) and it can be easily parallelized. It can even be a one-liner, but that isn't very readable.
At least the PostScript variant used in PDF 1.7 only supports straight lines and cubic Bézier curves as path segments [0]. You may be able to hack together true circular disks using round line caps and joins.
Heh, this would have been super helpful last summer as were nearing completion on the construction of our new home. We have an elliptical arch in our front portico that the carpenter was having trouble getting right...
I'm a little surprised that your architect didn't include layout information for the carpenters in their drawings. The whole point of the drawings is to enable the builder to faithfully reproduce the design, after all.
I'm not sure if it applies to the original commenter's situation but in many places, especially in the US, you don't need an architect to build a new home beyond rubber stamping some documents on file with the town, and sometimes that's not even required.
It's one of the reasons a lot of residential development, especially suburban development of the last 40 years, looks as bad as it does. Little to no architectural thought goes into many new homes beyond what's easiest to build.
I was kind of hoping that if you have an elliptical (or false elliptical) arch that there was an architect involved at some point.
I grew up in an area with a lot of corn field subdivisions and McMansions. They tend to have a lot of volume/floor space and incredibly poor detailing. On the one hand, I find it hard to believe that anyone building such a big, cheap (at least in the details sense) home on spec would include an elliptical arch. On the other hand, the kinds of contractors that put them up would likely be the kind that struggled to execute such a design element.
I say this having just been very humbled putting up crown in a bathroom. People tell me it looks great, and I reply that it had sure better considering the two weeks of evenings that went into putting it up!
I've found that absurd, gibberish rooflines, and weird, uneven jutting-out bits everywhere, plus garages shoved way out in front (?WHY?) do make the houses look bigger, which may be desirable for a builder.
I've shopped for houses a lot, and after "training" mostly on typical '80s+ suburban houses, noticed that when looking at older houses with saner, calmer designs, I'd have to add 500-1,000(!)sqft to my first-impression guesses at their size to get close, while I'd gotten pretty good at guessing the "McMansion" and mini-McMansion style. The older designs don't look as big, at the same size.
Don't these ovals suffer from the problems of g1 continuity, like for example ugly reflections if such an oval is made into a 3D reflective object? The reflections getting suddenly tighter where arcs connect. Or is it not noticeable if enough arcs are used?
Why is it that a constant curvature is preferable for a builder? Is it that construction forms etc are fabricated basically by compass and straightedge still?
As the author puts it [0], the issue is that if you expand or contract an ellipse by a fixed radius in every direction (e.g., if you create inner and outer walls for an elliptical corridor of constant width), then the resulting curve is no longer an ellipse. In contrast, piecewise-circular arcs can be expanded or contracted and remain piecewise-circular, though the control points will shift around as you transform it.
If you want the resulting geometry to be precise you are pretty much limited to equivalent of compass and straightedge (or well, in mechanical engineering lathe and mill, which are basically the same things). if you start somehow calculating the geometry you are limited by precision of the calculation, but more importantly by precision of measuring distances you can achieve.
Did I miss where an actual oval is explained in this method? I assume it is just drawing the same arc upside down but wanted to be sure there isn’t a follow up post or something.
Yeah, they're symmetrical. You use the three-center method doing isometric drawings by hand all the time. The second side is a mirror image of the first.
The four-centered arch is another one that you see a lot of
https://en.m.wikipedia.org/wiki/Four-centred_arch
Interesting contrast to this is that most modern vector 2D graphics API use PostScript/PDF drawing model that cannot represent exact circles and elipses and these are instead interpolated by a bunch of bezier curve segments.
Bezier curves (the regular kind) cannot represent circles and ellipses, this is true. However, rational Bezier curves can [1]. I don't know if PostScript has a rational Bezier primitive, but some drawing libraries do. A prominent example of a library that does is Skia, the rendering engine behind Chrome and Android.
[1] - e.g. https://en.wikipedia.org/wiki/B%C3%A9zier_curve#Rational_B%C...
I don't think that's true. Drawing Bezier curves is not any easier than drawing ellipses, and in fact I'm pretty sure it's even more difficult. If someone wants to approximate an ellipse the easiest solution is to just draw a bunch of short segments.
Can you provide examples? I have not seen an API not to support circles and arcs. SVG is AFAIK the most common API, you have there circles, arcs, ellipsis, quadratic and cubic beziers etc.
This is how SVG and most other drawing programs are implemented under the hood.
Down at the lowest level the engine can only really do straight lines. You may ask for a circle or a bezier, but the engine only approximates this with enough fidelity that you wouldn't notice the flaws. Beziers end up being approximated as a bunch of tiny line segments. Circles and ellipses can be approximated pretty easily as a handful of beziers.
Beziers are particularly useful because they have some handy properties. You don't need to do any trig to calculate them (like you do for arcs). They're extremely simple to compute, and any bezier curve can be split into two smaller bezier curves. That last fact makes it easier to have varying fidelity along the curve.
Source: was on the original SVG Working Group team @ Adobe.
FWIW, here's the bezier algorithm in Python:
You can pass it any order of bezier (cubic, quadratic, etc.) for any number of dimensions, and how for along the interpolation you are. It will give you a single point. Connect a bunch together to get your curve. You can actually stop when `len(handles) == 2` and you get line segments instead of points, but it's slightly less accurate.You can see the math for computing a point (or segment) is trivial (addition and multiplication) and it can be easily parallelized. It can even be a one-liner, but that isn't very readable.
Here's a good website for visualizing the math: https://www.summbit.com/blog/bezier-curve-guide/
At least the PostScript variant used in PDF 1.7 only supports straight lines and cubic Bézier curves as path segments [0]. You may be able to hack together true circular disks using round line caps and joins.
[0] https://opensource.adobe.com/dc-acrobat-sdk-docs/pdfstandard..., sec. 8.5.2, "Path Construction Operators"
Heh, this would have been super helpful last summer as were nearing completion on the construction of our new home. We have an elliptical arch in our front portico that the carpenter was having trouble getting right...
I'm a little surprised that your architect didn't include layout information for the carpenters in their drawings. The whole point of the drawings is to enable the builder to faithfully reproduce the design, after all.
I'm not sure if it applies to the original commenter's situation but in many places, especially in the US, you don't need an architect to build a new home beyond rubber stamping some documents on file with the town, and sometimes that's not even required.
It's one of the reasons a lot of residential development, especially suburban development of the last 40 years, looks as bad as it does. Little to no architectural thought goes into many new homes beyond what's easiest to build.
I was kind of hoping that if you have an elliptical (or false elliptical) arch that there was an architect involved at some point.
I grew up in an area with a lot of corn field subdivisions and McMansions. They tend to have a lot of volume/floor space and incredibly poor detailing. On the one hand, I find it hard to believe that anyone building such a big, cheap (at least in the details sense) home on spec would include an elliptical arch. On the other hand, the kinds of contractors that put them up would likely be the kind that struggled to execute such a design element.
I say this having just been very humbled putting up crown in a bathroom. People tell me it looks great, and I reply that it had sure better considering the two weeks of evenings that went into putting it up!
I've found that absurd, gibberish rooflines, and weird, uneven jutting-out bits everywhere, plus garages shoved way out in front (?WHY?) do make the houses look bigger, which may be desirable for a builder.
I've shopped for houses a lot, and after "training" mostly on typical '80s+ suburban houses, noticed that when looking at older houses with saner, calmer designs, I'd have to add 500-1,000(!)sqft to my first-impression guesses at their size to get close, while I'd gotten pretty good at guessing the "McMansion" and mini-McMansion style. The older designs don't look as big, at the same size.
Don't these ovals suffer from the problems of g1 continuity, like for example ugly reflections if such an oval is made into a 3D reflective object? The reflections getting suddenly tighter where arcs connect. Or is it not noticeable if enough arcs are used?
Why is it that a constant curvature is preferable for a builder? Is it that construction forms etc are fabricated basically by compass and straightedge still?
As the author puts it [0], the issue is that if you expand or contract an ellipse by a fixed radius in every direction (e.g., if you create inner and outer walls for an elliptical corridor of constant width), then the resulting curve is no longer an ellipse. In contrast, piecewise-circular arcs can be expanded or contracted and remain piecewise-circular, though the control points will shift around as you transform it.
[0] https://medium.com/@brunopostle/hey-ovals-are-better-than-el...
If you want the resulting geometry to be precise you are pretty much limited to equivalent of compass and straightedge (or well, in mechanical engineering lathe and mill, which are basically the same things). if you start somehow calculating the geometry you are limited by precision of the calculation, but more importantly by precision of measuring distances you can achieve.
Did I miss where an actual oval is explained in this method? I assume it is just drawing the same arc upside down but wanted to be sure there isn’t a follow up post or something.
Yeah, they're symmetrical. You use the three-center method doing isometric drawings by hand all the time. The second side is a mirror image of the first.