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In relation to my activity as a teacher of Mathematics and Physics at an Art School, I would like how to connect in graphics programs, any arguments of the analytic geometry or trigonometry in particular with Adobe Illustrator.

For example if I have the equation of a circle of center in O(0,0) and radius 1:

x^2+y^2=1

is there a way to use this equation within a graphical program? Into my school we can use only olds and obsolete programs of Adobe Photoshop, Adobe Illustrator (CS3 or CS4?) on Mac.

Thank you all in advance.

PS: I know Inkscape, TikZ/PGF, Gimp, Gnuplot and other programs that they use the maths or physics.

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    You might want to read this on general subject, Generally useful for math inclied, lot of construction kind of features have use from math, as do programming. – joojaa Feb 2 at 13:37
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    Some other examples like joining – joojaa Feb 2 at 13:58
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    In Inkscape, use Extensions > Render > Function plotter, or one of these could be helpful: inkscape.org/gallery/=extension/?tags=render (be careful to read the installation instructions) – Moini Feb 3 at 14:58
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    I removed my comments, your answers to them look out now obscure. I'll insert an answer, if I get together something that's not already said here by others. The case is interesting. – user287001 Feb 3 at 16:16
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    Inkscape 0.92.4 does no longer work on XP, that's correct. 0.92.3 should still work, though. 0.92.2 is the last packaged version that is available for Mac, currently. All versions are available from the inkscape.org website. – Moini Feb 4 at 12:50
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Math is needed in the infrastructure that makes graphic work possible. One must calculate material consumption, space needs, lengths, volumes, weights, money, the need of memory megabytes etc.... And of course, the tools often have needed a lot of engineering work with complex math. That work belongs to engineers and mathematicians, artists use the results.

But graphic work itself? At least there's some generally accepted and useful composition and projection rules. You can show how to utilize Illustrator when using them and prove that the method is mathematically right.

If you plan to create drawings of organic shapes with math, that's possible, but the needed math obviously is far beyond the elementaries. Of course nothing prevents us to use the results. Clever programmers have generated for example terrain, hair, fur, skin and floral pattern generators.

If you do admit that this is artificial, you can show how quite rich patterns can be generated by varying parameters in function expressions. Illustrator's blending is one practical application of it. Actually the whole functionality of Illustrator or other graphical programs is based on math. That math can be presented, but mainly as qualitative and with minimalistic full examples. If you find a way to model something real with understandable math, that's great. You can try hair as blendig, but the shapes must be quite complex and it's impossible to present them as simple equations. The blending itself probably is presentable.

Here's an old case:

How to create this pattern?

This is elementary, but it is not possible to solve without elementary math.

Analyzing a photo and finding from there actual relations is one possible subject. For ex. reconstruct an object as seen from another direction or find some actual lengths and placements.

If someone is forced calculate something during making a drawing with a program, it's easily seen as a fault. So, do not create situations where creative process is interrupted with seemingly arbitary limitations where something must be calculated in the middle. I'm sure that artists suffer if they are forced to interrupt and perform something which they already have placed to a lower level.

If you find a way to prove that doing some math in the start can prevent a stalemate in the middle of a work, that surely is appreciated. Unfortunately I am not sophisticated enough that I could write proper and interesting examples.

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    You can allways point out to infinite number of arbitrary limitations but you also need math when drawing certain stuff by hand. – joojaa Feb 3 at 17:59
  • @joojaa Any other suggestion are are very welcome. – Sebastiano Feb 4 at 22:39
  • user287001 If it is possible to add at your answer an animation of the explanation or other details they are also very welcome. I have accepted your answer. – Sebastiano Feb 4 at 22:41
  • @Sebastiano My writings are general ideas they should be refined to actual lessons. After it they can be animated. I'm sorry, but you are the artist. I'm only a leisure time visionary with too light background and skill level for such job. You are the only here who knows something actual of the level of your students and what math studies there actually reach. Hopefully they should support art skill studies. – user287001 Feb 4 at 22:52
  • @user287001 Do you thing that my answer is bad or good? – Sebastiano Feb 10 at 20:02
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Illustrator (or InkScape) use 3rd-degree Bézier curves (3rd-degree parametric polynomials)(and strangely you cannot use them to make a perfect circle, even if you can approximate one rather easily). These curves can be defined by 4 points and the relationship between these points and the curve is rather intuitive, which is why they are used in drawing programs.

Their most interesting property is that they are "compatible" with affine transforms. The curve you obtain by rendering the curve again after applying the transform to just the anchors is the same that you would obtain by applying the transform to an infinity of points of the initial curve. This is especially useful because you can scale them at will...

On the other hand there are seemingly simple problems that cannot be solved analytically such as computing the length of a curve, finding the intersection of two curves, or defining a curve which is at a constant distance of another (but there are algorithms to compute usable approximations). In other words to use them efficiently you have to be a rather pragmatic mathematician :)

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    Ive often wondered why we do not spline basis that would have the property that ist length can be computed analytically. This would have many useful implications, but if you would udr rational bezier splines then we could have circles, engineers do use rational b-splines so thay can make conics. (oddly they never seem to use it for other than conics), OTOH i also wonder why graphic designers dont us b-splines since they would make many of their day to day tasks easier. – joojaa Feb 6 at 20:21
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    But then again discrete math is still math, being able to use derivates or fourier on images is a good tool. Anyway, the reason to be careful when to introducing math to graphic design students is simply that theres too many people who have been traumatized by school mathematics among graphic designer students. There's not much you can do for them. – joojaa Feb 6 at 20:24
  • @joojaa I like to be and have always been a person of heart and how we can all make mistakes. I have to admit that my students are not traumatized, on the contrary they are always proactive and they want to learn and appreciate my lessons based on models and realities very much. I smiled now when you wrote the term traumatize. Surely it's just my opinion but I think that it's the many artists who teach in art schools who are not able to make people appreciate mathematics in drawing. – Sebastiano Feb 6 at 21:00
  • @joojaa In fact, my question is not clearly voted because for many people mathematics and physics are just marginal disciplines. Today I have justified, keeping in mind your answers, some mathematical aspects of InkScape and Krita software. Any further comments and answers are welcome. – Sebastiano Feb 6 at 21:00
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    @Sebastiano I am not a graphic designer i am a scientist and engineer, but i choose to teach engineers and artists. For me math is a tool, a useful tool still just a tool. I know a lot of math, but a lot of useful tools are still outside my day to day need. For example i find some areas of control theoy a bit tedious. – joojaa Feb 6 at 22:08
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...is there a way to use this equation within a graphical program?

It sounds like you are asking if and how it is possible for a user of mainstream graphics programs to use math (especially trig) expressions as input for creating or manipulating graphics. If so:

Adobe Illustrator provides very little in that regard in its standard interface. For example, its value fields will accept basic math operators (+, -, *, /), but only of one kind (addition and subtraction or multiplication and division) in a keyed-in expression.

However, Illustrator provides its own API and scripting model for Javascript, AppleScript, and Visual Basic for Applications, with complete documentation. So you (or your students) can build quite elaborate math-driven graphics "features" for use in the program.

Inkscape is similarly scriptable using Python.

Corel's CorelDRAW (and Technical Designer) provide for application scripting via VBA and VSTA, with editing support within the standard interface.

Other mainstream 2D vector drawing programs provide far more math-driven features or support for expressions in value fields in their standard interfaces. Examples:

Canvas GFX, Inc's Canvas provides its Math Expression 2D feature for creating paths based on Cartesian and Polar expressions.

Serif's Affinity Designer supports expressions including trig functions directly in the value fields of its Transform palette. Its Grids and Axis Manager feature includes support for user-customizable drawing grids, with both customizable axonometric presets and fully user-defined obliques; a good platform by which math students could employ trig toward immediate practical graphics advantage.

  • +1 Jet thank you very much for your collaboration. I use InkScape and Adobe Illustrator with LaTeX. Can you provide to explain your answer putting also the picture to have a complete answer? – Sebastiano Feb 10 at 20:01

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