### Re-inventing the wheel

Firstly, massive thanks are in order for such a storming positive reception of this new site – I really hope this will become a good resource for everyone interested in the fields these libs are addressing and I will try to update everyone (and everything) as often as I can. In the near future that will include a bit of future gazing, but today I’ve been re-inventing the wheel, literally…

I’ve been working on some designs for custom made wheels for a potential installation project. The aim is to lasercut some light-weight & good looking acrylic wheels and so I’ve been literally searching for a few candidate solutions, which allow me to cut out large pieces of material from inside the wheel to keep the weight down whilst still being sturdy enough (thankfully the load on them will be minimal so I can skip the structural analysis).

Not wanting to do this exploration/hike manually, I opted for the Processing PDF output route and have written this little tool below to show me my options (at least some of them) and also tweak them easily.

Apart from Processing, the tool also makes use of these 3 classes from the toxiclibscore package:

## Vec2D

This class, alongside its sibling Vec3D, is likely one of the most used classes of the entire library. Since 1.0 Processing has its own vector class too now, but Vec2D/3D are far more feature complete, implement a fluent interface for more legible code when dealing with complex vector maths and *can* be faster too (especially for 2D, but also by helping you to avoid temporary object creation). Furthermore Vec2D also has support for polar coordinates (Vec3D has spherical as equivalent) which was very helpful for building the tool below:

Normally, you’d specify a Cartesian point with:

Vec2D v = new Vec2D(x, y);

However, if we want to interpret our vector as polar coordinates, we’re using the x component to specify the radius and the y component as rotation angle. To convert the vector back into Cartesian space (e.g. our screen) we can use the `.toCartesian()`

method…

Vec2D v = new Vec2D(radius, theta).toCartesian();

In a similar manner, you can also transform a Cartesian point into polar coordinates:

// here we 1st create a copy of v and then convert that one Vec2D p = v.copy().toPolar();

To see this basic usage pattern in more context, take a closer look at the `drawHoles(`

) method below (lines 114-134) or check out the PolarUnravel demo bundled with the core lib.

## Spline2D

Each cut-out shape in the wheel is created from a simple spline shape, specified using only 4 points. As mentioned above, these anchor points are specified using (initially) polar coordinates and the Spline2D class is then computing control points (handles) for each of these given points automatically. As user you can also specify tightness & subdivisions for the computed curve vertices in between. While the lack of direct manual control over the spline handles might be a shortcoming in some situations, I generally found it easier to work with this automated version, especially when working with long curves and not only single bezier segments.

The next release of the library will also add a decimator method to this class which will enable the sampling of the curve at a uniform interval (i.e. all successive points of the returned list have the same distance (within a tolerance), regardless of curvature). This feature comes from the ParticlePath class briefly described in the Happy 2010 post.

Computing symmetrical handles for the curve endpoints is another still outstanding feature, but currently low priority (unless someone has got a patch ready for that ;)

The next image shows the 4 anchor points of each spline shape in green and all other computed vertices in pink.

## UnitTranslator

This class is the sole, lonely member of the `toxi.math.conversion`

package, but is especially useful for those of you using Processing for digital fabrication or creating printed outputs. It provides the following conversion methods, making it trivial to e.g produce PDF outputs at the right physical dimensions (without trial & error):

`millisToPixels(double mm, int dpi)`

– Converts millimeters into pixels.`millisToPoints(double mm)`

– Converts millimeters into PostScript points.`pixelsToInch(int pix, int dpi)`

– Converts pixels into inches.`pixelsToMillis(int pix, int dpi)`

– Converts pixels into millimeters.`pixelsToPoints(int pix, int dpi)`

– Converts pixels into points.`pointsToMillis(double pt)`

– Converts points into millimeters.`pointsToPixels(double pt, int dpi)`

– Converts points into pixels.

In the demo below, we’re using it to specifiy the wheel radius and drill holes in mm and then automatically calculate the required window size in pixels. In that case we’re however not using `millisToPixels()`

, but `millisToPoints()`

because PDF units are in points which are interpreted as 1 pixel (at 72 dpi)…

## The Wheel tool

Just copy & paste the code below into Processing and hit *Run* to fill up your hard disk (kidding, by default it only generates 120 different small PDF files/variations). The variations are created by iterating over all permutations of 4 parameters: number of symmetry steps, inset radius, core radius, alternate core radius.

/** * Generative wheel designs utilizing Vec2D polar coordinates, * splines and unit conversion. By default each design is exported as * PDF & PNG file, but can be turned off by setting the doExport flag to false. * * Usage: if export is enabled simply run & wait until all permutations * have been generated. Else press 'x' to activate next permutation or * use - / = to adjust the ARC_WIDTH parameter which defines the * size of the cutouts. * * (c) 2010 Karsten Schmidt, PostSpectacular Ltd. * * More info: http://toxiclibs.org/2010/01/re-inventing-the-wheel/ * Source code licensed under GPL v3.0. See http://www.gnu.org/licenses/gpl.html */ import processing.pdf.*; import toxi.geom.*; import toxi.math.conversion.*; // bleed in mm int BLEED = 6; // wheel radii & document size float R = (float)UnitTranslator.millisToPoints(370 / 2); float W = 2 * R + 2 * (float)UnitTranslator.millisToPoints(BLEED); float EMPTY_CORE_SIZE = (float)UnitTranslator.millisToPoints(5); // start number of main segments int NUM_SEGMENTS = 3; // normalized parameters float INSET_RADIUS = 0.8f; float OFFSET = 0.1f; float CORE_RADIUS = 0.15f; float ALTCORE_RADIUS = 0.3f; float ARC_WIDTH = 0.4f; // optional mounting holes int NUM_DOTS = 18; float DOT_RADIUS = 0.97f; float DOTSIZE = (float)UnitTranslator.millisToPoints(2); // number of subdivisions for spline vertex computation int SPLINE_SUBDIV = 20; // flag for PDF & PNG export of all permutations boolean doExport = true; public void setup() { size((int) W, (int) W); // turn off automatic redraws when not exporting if (!doExport) { noLoop(); } } public void draw() { String frameID = "wheel-" + NUM_SEGMENTS + "-" + nf(INSET_RADIUS, 1, 2) + "-" + nf(ALTCORE_RADIUS, 1, 2) + "-" + nf(CORE_RADIUS, 1, 2) + "-" + nf(ARC_WIDTH, 1, 2); println(frameID); if (doExport) { beginRecord(PDF, "out/" + frameID + ".pdf"); } background(255); noStroke(); fill(0); translate(width / 2, height / 2); ellipseMode(RADIUS); ellipse(0, 0, R, R); fill(255); ellipse(0, 0, EMPTY_CORE_SIZE, EMPTY_CORE_SIZE); // main holes drawHoles((INSET_RADIUS + OFFSET) * R, CORE_RADIUS * R, (INSET_RADIUS + OFFSET + CORE_RADIUS) / 2 * R, ARC_WIDTH, 0, NUM_SEGMENTS); if (CORE_RADIUS >= 0.5) { drawHoles(CORE_RADIUS * 0.85f * R, 0.2f * R, (CORE_RADIUS * 0.85f + 0.2f) / 2 * R, ARC_WIDTH, 0, NUM_SEGMENTS); } // small cutouts drawHoles(INSET_RADIUS * R, ALTCORE_RADIUS * R, (INSET_RADIUS + ALTCORE_RADIUS) / 2 * R, ARC_WIDTH / 2, PI / NUM_SEGMENTS, NUM_SEGMENTS); if (ALTCORE_RADIUS >= 0.5) { drawHoles(ALTCORE_RADIUS * 0.85f * R, 0.3f * R, (ALTCORE_RADIUS * 0.85f + 0.3f) / 2 * R, ARC_WIDTH / 2, PI / NUM_SEGMENTS, NUM_SEGMENTS); } // drill holes drawDots(NUM_DOTS, DOT_RADIUS * R, DOTSIZE); if (doExport) { endRecord(); saveFrame("png/" + frameID + ".png"); nextPermutation(); } } void drawDots(int num, float radius, float s) { float delta = TWO_PI / num; for (int i = 0; i < num; i++) { Vec2D p = new Vec2D(radius, i * delta).toCartesian(); ellipse(p.x, p.y, s, s); } } void drawHoles(float outerR, float innerR, float centerR, float radiusWidth, float thetaOffset, int num) { radiusWidth *= PI / num; Spline2D s = new Spline2D(); // define point in polar coordinates, then convert them Vec2D p = new Vec2D(outerR, 0).toCartesian(); Vec2D a = new Vec2D(outerR, radiusWidth).toCartesian(); Vec2D b = new Vec2D(centerR, radiusWidth).toCartesian(); Vec2D c = new Vec2D(innerR, 0).toCartesian(); Vec2D d = new Vec2D(centerR, -radiusWidth).toCartesian(); Vec2D e = new Vec2D(outerR, -radiusWidth).toCartesian(); // add points to spline & compute s.add(p).add(a).add(b).add(c).add(d).add(e).add(p); java.util.List verts = s.computeVertices(SPLINE_SUBDIV); float delta = TWO_PI / num; for (int i = 0; i < num; i++) { pushMatrix(); rotate(i * delta + thetaOffset); drawPath(verts); popMatrix(); } } void drawPath(java.util.List verts) { beginShape(); for (Iterator i = verts.iterator(); i.hasNext();) { Vec2D v = (Vec2D) i.next(); vertex(v.x, v.y); } endShape(); } public void keyPressed() { if (key == 'x') { nextPermutation(); } else if (key == '-') { ARC_WIDTH -= 0.02; } else if (key == '=') { ARC_WIDTH += 0.02; } redraw(); } void nextPermutation() { CORE_RADIUS += 0.2; if (CORE_RADIUS >= INSET_RADIUS) { CORE_RADIUS = 0.15f; ALTCORE_RADIUS += 0.2f; if (ALTCORE_RADIUS >= INSET_RADIUS) { ALTCORE_RADIUS = 0.3f; INSET_RADIUS += 0.1f; if (INSET_RADIUS > 0.8) { NUM_SEGMENTS++; INSET_RADIUS = 0.8f; if (NUM_SEGMENTS > 12) { exit(); } } } } }

Finally, below are some more images of variations created with this tool (see the full size & set on flickr):