FE 'tkGooie' Utilities
'IMAGEcreators - Shaded3D' group
A Tcl-Tk GUI to
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This 'draw_3colorGradient_ ISOSCELES-TRIANGLE_ colorShaded3Dedges'
I am interested in making nice images for 'toolchest' and 'drawer' backgrounds (and other GUI embellishments), as I have indicated at wiki.tcl.tk (and on this freedomenv.com site) --- as indicated by
In 2012, I made at least 4 scripts for making objects with shaded-edges, yielding a 3D-look. I posted those scripts at wiki.tcl.tk, and on this freedomenv.com site, in web pages:
These scripts used the concept of a 'color-shading-metric' --- a scalar function defined over the x,y pixels of a Tk canvas image. The metric had the value 1.0 on the boundary of the shape, and the value 0.0 in the center.
The metrics that I used for the first 3 shapes above were
(See the page-links above for details of the 'shading-metrics' for each.)
Those utilities worked with 2 colors --- blending the two colors --- from one color 'in the middle' to a 2nd color 'at the edge'.
I have had it on my things-to-do list (for about a year now) to explore using 'barymetric coordinates' to blend THREE colors --- assigned to the 3 vertices of a triangle --- over the interior of the triangle.
This interest in gradiating 3 colors over a triangle comes partly from my creating some scripts for displaying 3D surfaces --- and for reading and displaying 3D model files --- such as the Tk scripts that I posted at wiki.tcl.tk, and on this freedomenv.com site, at web pages:
In 3D modeling, to get nice rendering of colors over a triangular (or quadrilateral) mesh, it is common practice to use 'Gouraud' shading (simpler than Phong shading) to get an improved appearance of the model.
Gouraud shading works off of colors at the vertices of the polygons. This color shading on the surfaces of 3D models has motivated me to explore using barymetric/barycentric coordinates to get Gouraud-like shading across a triangle.
Adding 'Edge Shading':
At the same time as doing this '3 color blend', I wanted to use the 'shading-metric' technique (like I used in the scripts to make 3D-like rectangles, super-ellipses, super-formula shapes, and donut shapes) to put a 'shaded edge' on the color-gradient triangle.
If the results turned out nicely, I thought I might find some use for the '3-shaded-colors-with-shaded-edges' triangles in making 'embellished' Tk GUI's.
Barymetric math for other projects:
In any case, I will have developed the code for dealing with calculating barymetric coordinates for points within triangles.
That math (and Tcl-Tk code) will have application to some other projects on my things-to-do list --- image-warping/morphing projects and --- maybe ---- 3D modeling projects.
Some Images To Tell The Story
The code for doing the combination of
takes quite a lot to describe. For that description, I refer you to comments in the code below. And I present an overview in a couple of sections near the bottom of this page.
I skip the details for now and show some images of the GUI that I created to do the 'barymetric/barycentric blending' and 'edge shading' outlined above.
Here is an image that shows what the 3-color-isosceles-triangle-making GUI (with barymetric-color-blending and edge-shading) looks like when it first comes up.
Note that there are 3 color buttons across the top of the GUI for assigning a color to each of the vertices of the triangle. And there is a color button for assigning a background color --- which is also the color that this utility uses to create a 'shaded edge' on the triangle.
Also note that there is a 'scale' widget on the GUI that allows the user to set an exponent, N, that controls what I call 'extensity' --- the extent and intensity of the shading that occurs as one approaches the edges of the triangle.
The following image shows that when N is at an extreme value (far from 1), there is very little apparent shading at the edges of the triangle. In fact, you can see the 'jaggies' of the edges of the triangle meeting the black background.
You can set N so that there is 'strong' background-color blending at the edges of the triangle. That is, the background color is strongly weighted in blending the background color with any pixel color of the triangle. (The pixel colors in the interior of the triangle are determined by a 'barymetric blend' of the 3 vertex colors of the triangle.)
To 'turn things updside down, color-wise', I changed the vertex colors from CMY colors to RGB colors --- and I changed the background color from black to white. Here is the result.
The size and aspect-ratio of the triangle can be changed by resizing the window and clicking on the 'ReDraw' button. The new size of the canvas widget is used to determine the new locations of the vertices of the isosceles triangle.
Note that the color buttons call on a color-selector-GUI script to set those colors.
You can make that color-selector script by cutting-and-pasting the code from the page that offers a non-obfuscated color selector GUI on this site.
DESCRIPTION OF THE CODE
Below is the code that produced this GUI.
There are comments at the top of the code, in a section titled 'USING THE GENERATED IMAGE', that describe how one could make use of images produced by this GUI.
I follow my usual 'canonical' structure for Tk code, for this Tk script:
0) Set general window & widget parms (win-name, win-position, win-color-scheme, fonts, widget-geometry-parms, win-size-control, text-array-for-labels-etc). 1a) Define ALL frames (and sub-frames, if any). 1b) Pack ALL frames and sub-frames. 2) Define & pack all widgets in the frames, frame by frame. Within each frame, define ALL the widgets. Then pack the widgets. 3) Define keyboard and mouse/touchpad/touch-sensitive-screen 'event' BINDINGS, if needed. 4) Define PROCS, if needed. 5) Additional GUI initialization (typically with one or more of the procs), if needed.
This Tk coding structure is discussed in more detail on the page A Canonical Structure for Tk Code --- and variations.
This structure makes it easy for me to find code sections --- while generating and testing a Tk script, and when looking for code snippets to include in other scripts (code re-use).
I call your attention to step-zero. One new thing that I have started doing recently is using a text-array for text in labels, buttons, and other widgets in the GUI. This can make it easier for people to internationalize my scripts. I will be using a text-array like this in most of my scripts in the future.
Experimenting with the GUI
As in all my scripts that use the 'pack' geometry manager (which is all of my 100-plus Tk scripts, so far), I provide the four main pack parameters --- '-side', '-anchor', '-fill', and '-expand' --- on all the 'pack' commands for the frames and widgets.
That helps me when I am initially testing the behavior of a GUI (the various widgets within it) as I resize the main window.
I think I have found a good setting of the '-side', '-anchor', '-fill', and '-expand' parameters on the 'pack' commands for the various widgets of this GUI. In particular ...
The 'canvas' widget expands/contracts appropriately when the window size is changed --- and button and label widgets stay fixed in size and relative-location as the window size is changed.
If anyone wants to change the way the GUI configures itself as the main window size is changed, they can experiment with the '-side', '-anchor', '-fill', and '-expand' parameters on the 'pack' commands for the various widgets --- to get the widget behavior that they want.
Note: I have found that one can get some quite surprising self-changing movement of the slider-button on scale widgets if you allow a scale widget to expand and allow the window to expand. So you may want the scale widgets to stay fixed in length ('-fill none') and/or use '-expand 0' when packing scale widgets --- rather than using '-fill x -expand 1' with scale widgets and an expandable GUI window.
Additional experimentation: You might want to change the fonts used for the various GUI widgets. For example, you could change '-weight' from 'bold' to 'normal' --- or '-slant' from 'roman' to 'italic'. Or change font families.
In fact, you may NEED to change the font families, because the families I used may not be available on your computer --- and the default font that the 'wish' interpreter chooses may not be very pleasing.
I use variables to set geometry parameters of widgets --- parameters such as border-widths and padding. And I have included the '-relief' parameter on the definitions of frames and widgets. Feel free to experiment with those 'appearance' parameters as well.
Some features in the code
There are plenty of comments in the code to describe what most of the code-sections are doing.
See the top of the 'PROCS' section for a list of the procs used in this Tk script. See comments in the procs for details on the purpose of each proc and for details on the methods by which each proc was implemented.
Here is a quick overview of the procs --- to give an idea of the 'guts' of this utility:
'ReDraw' - called - in the 'Additional GUI Initialization' section at the bottom of this script, - by the 'set_*_color*' procs, - by a button1-release binding on the N-scale widget, - by a click on the 'ReDraw' button. 'set_vertex_color1' - called by the vertex1-color button 'set_vertex_color2' - called by the vertex2-color button 'set_vertex_color3' - called by the vertex3-color button 'set_background_color' - called by the background color button 'update_color_labels' - called by the color procs to update colors and text on the color buttons 'popup_msgVarWithScroll' - called by 'Help' button
The most complex code is in the 'ReDraw' proc. The code is quite complex --- to handle the math for
So, for now, I have NOT attempted to significantly reduce the number of 'put' commands used to draw an image. I am currently using a 'put' command for each pixel in the rectangular, in-memory 'photo' image structure.
Eventually, I plan to return to this code and, by using 'horizontal-scanlines' (of hex-colors), reduce the 'put' commands to either one 'put' per scanline --- or one 'put' for the entire image.
When using a 'put' of a hexcolor at each individual pixel, it takes about 1 to 10 seconds to draw the image --- depending on the size of the window (and thus the size of the canvas and the size of the 'photo' structure).
By using the 'horizontal-scanlines' technique with the 'put' command, I will probably be able to reduce the draw times to about half a second or less --- even for large images.
You can see the comments in the 'ReDraw' routine for details on the math and image building going on there.
Comments in the Code
It is my hope that the copious comments in the code will help Tcl-Tk coding 'newbies' get started in making GUI's like this. Without the comments --- especially in the 'ReDraw' proc --- the code would look like 'too much monkey business to be involved in', to quote Chuck Berry.
Without the comments, potential young Tcler's might be tempted to return to their iPhones and iPads and iPods --- to watch the Falkland Islands' Funniest Home Videos.
The Tcl-Tk Script CODE
Here is a link to CODE for the Tk script
With your web browser, you can 'right-click' on this link --- and in the menu that pops up, select an item like 'Save Link Target As ...' --- to save this file to your local computer.
Then you can rename the file to remove the '.txt' suffix. Make sure that you have execute permission set on the file --- in order to execute the script.
I would like to give thanks to 'ulis' (deceased in 2008, R.I.P.) whose super-ellipse script on the page titled 'Mathematics jewels' at wiki.tcl.tk indicated to me that we could get nicely shaded edges on geometric shapes --- using 'image create photo' and a Tk canvas.
In 2013, I noticed that the link to his 'jewels' image has gone dead. I decided to make an image from my copy of his script and post the image on his page to replace the dead link. He deserves his image to be shown, so that people can easily see what he wrought.
In fact, here is a copy of the image that led me to this journey with making Tk canvas images with '3D-looking shaded edges'.
Also --- thanks go to James Emery, who has some nice PDF's on math topics --- in particular, on barycentric coordinates --- at a www.stem2.org web site. Emery seems to be active in STEM (Science, Technology, Engineering, and Math) education activities in the Kansas City area of the United States.
MATH FOR THE BARYMETRIC BLEND
I have adapted the following presentation from a PDF file titled 'The Simplex and Barycentric Coordinates' by James Emery - Latest Edit: 8/30/2012
Below are formulas for computing the barymetric coordinates --- L1,L2,L3 --- of a point P relative to a triangle with vertices P1,P2,P3.
If all 3 barycentric coordinates are between 0 and 1, the point is inside the triangle.
The general computation to determine an interior point, requires 11 additions or subtractions, 6 multiplications, 2 divisions, and 3 comparisons. The computation may be done as follows.
We let P1=(x1,y1), P2=(x2,y2), P3=(x3,y3), P=(x,y). Let
a11 = x1 - x3 a21 = y1 - y3 a12 = x2 - x3 a22 = y2 - y3 and b1 = x - x3 b2 = y - y3 .
Let D be the determinant:
D = (a11 * a22) - (a21 * a12)
Then we have
L1 = ((b1 * a22) - (b2 * a12)) / D L2 = ((a11 * b2) - (a21 * b1)) / D L3 = 1 - (L1 + L2)
Emery provided C program code for the above calculations, but I assembled the Tcl-Tk code, in the script above, from this set of formulas.
DERIVATION OF A 'SHADING METRIC' FOR THE TRIANGLE :
In my other 'shading metric' scripts above, I devised a scalar 'color-shading-metric' --- 'v' --- at each pixel point x,y that is within the rectangle/super-ellipse/super-formula/donut shape --- where v is between 0 and 1. And the value of v goes to 1.0 at the boundary of these shapes --- and is 0.0 in the middle of the shape.
Then v and (1-v) are applied to a user-selected background color and a user-selected 'fill' color for any pixel within the shape --- to get a 'weighted-average' of the user-selected colors, for the shaded-color at a given pixel x,y.
For the triangle of the script on this page, we use the barycentric coordinates of any pixel within the triangle --- to formulate our 'shading metric'.
For a given pixel x,y, let us say its barymetric coordinates are (L1,L2,L3) where L1 + L2 + L3 = 1.
Note that at the edges (and vertices) of the triangle, at least one of the 3 barymetric coordinates is 0.
Let us say that we want our metric, v(x,y) = v(L1,L2,L3), to be 1.0 at the edges of the triangle. (This is helpful when we later use an 'extensity' exponent to make 'crisper' shading at the edges.)
Also, we want v to be a 0.0 at the barycenter of the triangle.
Note that the 'barymetric center' of the triangle is at the point/pixel where (L1,L2,L3) = (1/3,1/3,1/3). And when one looks at any point within the triangle which is not on that center, at least one of the coordinates is less than 1/3.
Note that the quantity u = 3.0 * min(L1,L2,L3) is 1.0 at the barycenter and is zero on the edges of the triangle. We want the opposite for our scalar 'color-shading-metric'.
Based on these observations, for our 'color-shading-metric', we will use
v(x,y) = v(L1,L2,L3) = 1.0 - ( 3.0 * min(L1,L2,L3) )
Note that v is 1.0 on the edges (and vertices) of the triangle, it is 0.0 at the barymetric center, and it is between 0 and 1 at other points in the triangle.
So we have a suitable metric, v.
USING THE SHADING-METRIC:
At a point x,y, we determine the 'shaded color' at the point by using a color interpolated between
1) Rbary,Gbary,Bbary --- 'color-bary', say --- which is the barymetrically-determined color of the point/pixel x,y calculated as follows: Let the RGB values of vertex 1, 2, and 3 be (R1,G1,B1) and (R2,G2,B2) and (R3,G3,B3). Then Rbary = (L1 * R1) + (L2 * R2) + (L3 * R3) Gbary = (L1 * G1) + (L2 * G2) + (L3 * G3) Bbary = (L1 * B1) + (L2 * B2) + (L3 * B3) where L1,L2,L3 are the barymetric coordinates of (x,y). and 2) a user-selected background color, 'color-bkgd', say.
We calculate the 'shaded color' at x,y by calculating a weighted average based on applying the factor v to 'color-bkgd' --- and applying (1 - v) to 'color-bary'. That is,
shaded-color = v * color-bkgd + (1 - v) * color-bary.
We actually calculate via formulas like
Rshaded = v * Rbkgd + (1 - v) * Rbary
Thus we will get an edge-shading (the 3D or 'soft edge' effect) for the triangle shape.
Actually, it turns out that v and 1-v give a rather washed-out (too gradual) shading effect. It is better if we raise v to a power N and use v^N and (1 - v^N), where N can be taken to be an integer greater than or equal to 1.
This is easy to do in Tcl-Tk, by using the 'pow' (power) function.
It turns out we get rather pleasing 'crisp', shaded edges on the triangle when N is between say 6 and 12.
SOME POSSIBLE ENHANCEMENTS:
** Performance Improvement
Such a list would probably be on the order of 8 Megabytes in size. In the old days, that would have been a no-no. But since users typically are dealing with over a Gigabyte of free memory on their computers, 8 Megabytes is less than one percent of that un-used memory. Doing a single 'put', rather than a 'put' for each pixel of the rectangular 'photo' structure, is high on my list of enhancements --- if I live to re-visit this code.
** Edge-Shading Improvement
** Non-Isosceles Triangles
I am not sure that spending time on this is worthwhile. If this utility proves to be more experimental than useful, then maybe it is better to put this kind of feature in a more general utility.
** More Re-Sizing Controls
I could allow for more precise controls, including control of the margin of background color around the triangle, but, again, I am not sure that spending time on this is worthwhile. Re-sizing the triangle by changing the window size is probably sufficient for most uses of this demo-and/or-utility.
I would like to give my usual thanks to Ousterhout and the maintainers of the 'wish' interpreter for making possible all these mathematical and graphical utilities and experiments (that I have done and that are on my 'to-do' list).
UPDATE 2016 Jan 07:
I have addressed several of the 'possible enhancements' listed above:
In the process, I added several 'scale' widgets to the GUI --- resulting in the GUI seen in the following image.
In particular, the following changes have been made to the Tk script:
The code that was available in a code-link above has been replaced with with the new code. See the comments in the code, and the code itself, for details.
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The code was created in 2013 --- and posted 2013 Sep 05 at http://wiki.tcl.tk/38676.
This FE web page was created 2014 May 10 ---
as a backup and alternative to the wiki.tcl.tk page.