TweetFollow Us on Twitter

GX Rigid Body Dragging
Volume Number:12
Issue Number:8
Column Tag:New Technologies

Dragging and Rigid-Body
Transformations

QD GX can make your Mac a swinging place

By Lawrence D’Oliveiro

Note: Source code files accompanying article are located on MacTech CD-ROM or source code disks.

QuickDraw GX offers some interesting new possibilities for interactive graphics. This article is a note on one of them: the idea of dragging an object about, and having it rotate to follow the drag, just like real objects tend to do when you pull or push them.

The code I’m presenting here is by no means finished; think of it as a rough sketch, a demonstration of the concept. It is written in Modula-2, and makes heavy use of my standard libraries, which you can find at ftp://ftphost.waikato.ac.nz/pub/ldo/MyModLib.hqx.

Feel free to use my code as a starting point for your own experiments.

Transformations in QuickDraw GX

QuickDraw GX supports 3-by-3 mappings, which are capable of applying any kind of linear transformation to the geometry of a GX shape. (A linear transformation is one that preserves the straightness of lines. Thus, a GX mapping cannot transform a straight line to a curve, or vice versa.)

Linear transformations can be broken down into various simple types: translations (changes of position), rotations, scaling, skewing and perspective. In fact, any arbitrary linear transformation can be considered to be built from components of these types.

There are various useful subsets of linear transformations: an affine transformation is one where any pair of parallel lines remains parallel after the transformation. GX’s perspective transformations clearly are not affine; thus, an affine transformation is one that has no perspective component.

An important subset of affine transformations is the set of rigid-body transformations: these are ones that preserve the distance between any two given points. Rigid-body transformations consist only of translations and rotations; this corresponds intuitively to our notions of real rigid bodies, which cannot be scaled or stretched in any way, though you can usually move them around and reorient them any way you like (unless they’re a lot bigger or heavier than you are).

Figure 1. As the cursor pulls the arrow to the right, the arrow swings into line with the direction of motion. These are successive screen shots from my program; the upper-left corner of each shot is the upper-left corner of the window.

Dragging a Rigid Body

Consider what happens when you try to move a real rigid body. The body has a center of mass; if you orient the direction of your force so that it passes exactly through this point, the object will change its position without rotating. But if you offset your force by any amount, you will produce a torque, and the object will change both its position and its orientation. (For example, place a book on a smooth surface such a desk, grasp it by just one corner, and pull in a straight line; the book, as it starts to move, also swings around to line up its center of mass with the direction of travel.)

Figure 1 shows this happening in my program.

In the following analysis, I’ll follow the rules of Aristotelian, rather than Newtonian, physics: objects have no inertia, but they are subject to friction, which acts through their center of mass. Thus, they stop moving as soon as you stop pushing them.

Consider a body with its center of mass at a point C. Say you apply a small force on it at a point F, sufficient to displace that point to a new position G close to F. The center of mass is in turn moved to a new position D (Figure 2).

Figure 2. Movements... Figure 3. ...and analysis

The force you apply can be split into two components: the component parallel to CF moves the object without rotating it (hence the new position of the center of mass, D, lies on this line), while the component perpendicular to this line exerts a pure rotational force on the object without moving the center of mass.

In Figure 2, q is the angle CFG and can have any value, while f is the angle of rotation FCG, and is assumed to be small (you’ll see why shortly).

Now draw a line P1P2 parallel to CF, and project the points C, D, F and G onto this line at the points Cp, Dp, Fp and Gp (Figure 3). Gt is the intersection between FFp and CG. In this case, q is greater than 90°, so the angle GFGt is q - 90° (I’ll leave the analysis of the case where q is less than 90° as an “exercise for the reader”, as they say). Since f is small, FGtG is close to 90°, so the ratio FGtFG is approximately cos q - 90°, which equals sin q. Now, since GFGt is a right angle, sin f equals FGtCF which becomes FG sin qCF when you apply the above approximation.

For larger movements, where the angle f might not be small, simply split the movement into lots of small steps with correspondingly small f. If you want a fancy-sounding term for this mathematical trick, it’s called differential calculus.

Translating This Into Code

The next step is to write some actual code based on this analysis. In the following, I’ll intermingle declarations and statements to suit the exposition, rather than the requirements of strict language syntax (in other words, I’ll be following the order in which code is usually written).

VAR
 LastMouse, ThisMouse : Point;
 ThisDelta : gxPoint;
 ShapeCenter : gxPoint;

LastMouse is the point F, while ThisMouse is the point G. ThisDelta is the displacement FG, computed as follows:

 ThisDelta.x := IntToFixed(ThisMouse.h - LastMouse.h);
 ThisDelta.y := IntToFixed(ThisMouse.v - LastMouse.v);

ShapeCenter is supposed to be the center of mass, or center of geometry, of the shape. GX provides a call, GXGetShapeCenter, that is supposed to return this in the coordinate system of the shape geometry itself. You could then put this through the shape mapping to get the center in “local coordinates”. Unfortunately, when I tried this, I got incorrect results for a complex picture shape (a QuickDraw GX bug?). So, to avoid this problem, my code takes ShapeCenter as the center of the shape’s bounding rectangle, rather than of its actual geometry:

GXGetShapeLocalBounds(TheShape, Bounds);
ShapeCenter.x := (Bounds.right + Bounds.left) DIV 2;
ShapeCenter.y := (Bounds.bottom + Bounds.top) DIV 2;

Next we perform the computation of the actual rotation angle, using good old floating-point numbers instead of fixed-point ones:

VAR
 DeltaX, DeltaY, Delta : LongReal;
 RadiusX, RadiusY, Radius : LongReal;
 DragAngle, RotationAngleSin : LongReal;
 RotationAngle : LongReal;

DeltaX and DeltaY are the x- and y-components of the displacement FG, while Delta is the magnitude of this displacement.

DeltaX := Fix2Double(ThisDelta.x);
DeltaY := Fix2Double(ThisDelta.y);
Delta := Sqrt(Squared(DeltaX) + Squared(DeltaY));

Similarly, RadiusX and RadiusY are the x- and y-components of thedistance CF, while Radius is the magnitude of this distance.

RadiusX := Fix2Double(IntToFixed(LastMouse.h) - ShapeCenter.x);
RadiusY := Fix2Double(IntToFixed(LastMouse.v) - ShapeCenter.y);
Radius := Sqrt(Squared(RadiusX) + Squared(RadiusY));

DragAngle is the angle q. I compute it here from the difference between the angles of the lines FG and CF (ArcTan2(x, y) returns the angle with tangent y/x in the appropriate quadrant, taking account of the signs of x and y):

DragAngle := ArcTan2(DeltaX, DeltaY) - ArcTan2(RadiusX, RadiusY);

RotationAngleSin is the sine of the angle f. DLimit is just a routine that constrains its first argument to within the specified limits (in this case, between -1 and 1). The need for this constraint will become apparent later.

RotationAngleSin := DLimit
 (
 Delta * Sin(DragAngle) / Radius,
 FLOATD(-1),
 FLOATD(1)
 );

And finally, we compute the angle f itself, converting from the floating-point radians that SANE operates in, to the fixed-point degrees that QuickDraw GX uses:

RotationAngle := 
 Double2Fix(ArcSin(RotationAngleSin) * FLOATD(180) / Pi())

Applying the transformation to the shape is pretty straightforward. Simply obtain the shape’s existing mapping:

VAR
 ShapeMapping : gxMapping;
...
GXGetShapeMapping(TheShape, ShapeMapping);

then apply the appropriate movement and rotation:

MoveMapping
  (
 (*@target :=*) ShapeMapping,
 (*hOffset :=*) ThisDelta.x,
 (*vOffset :=*) ThisDelta.y
  );
RotateMapping
  (
 (*@target :=*) ShapeMapping,
 (*angle :=*) RotationAngle,
 (*xCenter :=*) IntToFixed(ThisMouse.h),
 (*yCenter :=*) IntToFixed(ThisMouse.v)
  );
GXSetShapeMapping(TheShape, ShapeMapping)

Note that the center of rotation used in the RotateMapping call is the point G rather than C. This is all right, because the direction of rotation calculated by the code is actually the opposite of that in the analysis; thus the effect is the same.

Other Matters

There are several other aspects of the example code that I’ve glossed over so far.

To keep the size of the source code down, the program itself has absolutely the minimum user interface I felt I could get away with. It doesn’t even have any menus! When you start it up, it already has a shape loaded that you can try dragging about. You can bring in a different shape by dragging it into the window, from a Finder clipping file or another drag-aware application (I’ve provided a few sample clippings you can try). To quit the program, click the close box in the window.

An important issue is how to do off-screen rendering, so that your on-screen drawing doesn’t flicker. There is no direct equivalent to QuickDraw GWorlds in QuickDraw GX, but the GX SDK library code shows you how to create a much more powerful alternative: an offscreen graphics context that can optimize drawing simultaneously for multiple screens, rather than just the deepest one that your window happens to cross.

My actual program uses a routine I wrote called MoveSprite, which creates temporary offscreen structures every time you call it to move an object. It automatically sizes these structures to cover only the affected on-screen area (and doesn’t bother doing off-screen drawing if the old and new positions of the object don’t overlap). Thus, this routine is simpler to use than explicitly creating separate off-screen structures and reusing them for the duration of the drag, though it may be slower.

One feature of the code is that it doesn’t rotate the shape if you drag it at a point close to its center, or if you hold the command-key down. I figure this sort of feature could be useful in a “real” program, where the user might not always want the object to rotate.

One pitfall you always have to keep in mind when doing calculations on a computer is rounding error. In this case, repeated calls to RotateMapping can accumulate distortions in the shape geometry, since the mapping elements cannot be computed exactly. To get around this, you should keep separate track of the shape position and rotation angle, and recompute the shape mapping from these values each time, instead of applying an incremental rotation to the previous mapping. This will keep the distortion within fixed bounds, instead of letting it accumulate. My code doesn’t do this, though I must admit I have yet to notice any distortions in shapes after repeated drags.

Finally, I should own up to one important liberty I have taken with the mathematical analysis. Remember how I kept saying that the displacement and rotation angle were assumed to be small? In fact, the code will happily compute arbitrarily large values for the FG displacement, depending on how quickly you can move the mouse, and how long it takes your machine to redraw the shape in between checking the mouse position. This means that, for large mouse movements, the code is no longer strictly MC (Mathematically Correct). It’s also why I put in the DLimit call.

But then, I figure this is part of the fun of programming. For instance, I have found that I can drag a shape to a corner of the window, and leave the mouse absolutely stationary outside the window, while the shape continues to spin round and round in the corner. Is this a consequence of the code hitting a non-MC situation? I don’t know - you tell me!

 
AAPL
$116.47
Apple Inc.
+0.16
MSFT
$47.98
Microsoft Corpora
-0.72
GOOG
$537.50
Google Inc.
+2.67

MacTech Search:
Community Search:

Software Updates via MacUpdate

Cobook 3.0.7 - Intelligent address book....
Cobook Contacts is an intuitive, engaging address book. Solve the problem of contact management with Cobook Contacts and its simple interface and powerful syncing and integration possibilities.... Read more
StatsBar 1.9 - Monitor system processes...
StatsBar gives you a comprehensive and detailed analysis of the following areas of your Mac: CPU usage Memory usage Disk usage Network and bandwidth usage Battery power and health (MacBooks only)... Read more
Cyberduck 4.6 - FTP and SFTP browser. (F...
Cyberduck is a robust FTP/FTP-TLS/SFTP browser for the Mac whose lack of visual clutter and cleverly intuitive features make it easy to use. Support for external editors and system technologies such... Read more
Maya 2015 - Professional 3D modeling and...
Maya is an award-winning software and powerful, integrated 3D modeling, animation, visual effects, and rendering solution. Because Maya is based on an open architecture, all your work can be scripted... Read more
Evernote 6.0.1 - Create searchable notes...
Evernote allows you to easily capture information in any environment using whatever device or platform you find most convenient, and makes this information accessible and searchable at anytime, from... Read more
calibre 2.11 - Complete e-library manage...
Calibre is a complete e-book library manager. Organize your collection, convert your books to multiple formats, and sync with all of your devices. Let Calibre be your multi-tasking digital... Read more
Herald 5.0.1 - Notification plugin for M...
Note: Versions 2.1.3 (for OS X 10.7), 3.0.6 (for OS X 10.8), and 4.0.8 (for OS X 10.9) are no longer supported by the developer. Herald is a notification plugin for Mail.app, Apple's Mac OS X email... Read more
Firetask 3.7 - Innovative task managemen...
Firetask uniquely combines the advantages of classical priority-and-due-date-based task management with GTD. Stay focused and on top of your commitments - Firetask's "Today" view shows all relevant... Read more
TechTool Pro 7.0.6 - Hard drive and syst...
TechTool Pro is now 7, and this is the most advanced version of the acclaimed Macintosh troubleshooting utility created in its 20-year history. Micromat has redeveloped TechTool Pro 7 to be fully 64... Read more
PhotoDesk 3.0.1 - Instagram client for p...
PhotoDesk lets you view, like, comment, and download Instagram pictures/videos! (NO Uploads! / Image Posting! Instagram forbids that! AND you *need* an *existing* Instagram account). But you can do... Read more

Latest Forum Discussions

See All

Ubisoft Gives Everyone Two New Ways to E...
Ubisoft Gives Everyone Two New Ways to Earn In-Game Stuff for Far Cry 4 Posted by Jessica Fisher on November 21st, 2014 [ permalink ] | Read more »
Golfinity – Tips, Tricks, Strategies, an...
Dig this: Would you like to know what we thought of being an infinite golfer? Check out our Golfinity review! Golfinity offers unlimited ways to test your skills at golf. Here are a few ways to make sure your score doesn’t get too high and your... | Read more »
Dark Hearts, The Sequel to Haunting Meli...
Dark Hearts, The Sequel to Haunting Melissa, is Available Now Posted by Jessica Fisher on November 21st, 2014 [ permalink ] Universal App - Designed for iPhone and iPad | Read more »
Meowza! Toyze Brings Talking Tom to Life...
Meowza! | Read more »
Square Enix Announces New Tactical RPG f...
Square Enix Announces New Tactical RPG for Mobile, Heavenstrike Rivals. Posted by Jessica Fisher on November 21st, 2014 [ permalink ] With their epic stories and gorgeous graphics, | Read more »
Quest for Revenge (Games)
Quest for Revenge 1.0.0 Device: iOS Universal Category: Games Price: $4.99, Version: 1.0.0 (iTunes) Description: The great Kingdom of the west has fallen. The gods ignore the prayers of the desperate. A dark warlord has extinguished... | Read more »
Threadz is a New Writing Adventure for Y...
Threadz is a New Writing Adventure for You and Your Friends Posted by Jessica Fisher on November 21st, 2014 [ permalink ] In the tradition of round-robin storytelling, | Read more »
SteelSeries Stratus XL Hardware Review
Made by: SteelSeries Price: $59.99 Hardware/iOS Integration Rating: 4 out of 5 stars Usability Rating: 4.5 out of 5 stars Reuse Value Rating: 4.25 out of 5 stars Build Quality Rating: 4.5 out of 5 stars Overall Rating: 4.31 out of 5 stars | Read more »
ACDSee (Photography)
ACDSee 1.0.0 Device: iOS iPhone Category: Photography Price: $1.99, Version: 1.0.0 (iTunes) Description: Capture, perfect, and share your photos with ACDSee. The ACDSee iPhone app combines an innovative camera, a powerful photo... | Read more »
ProTube for YouTube (Entertainment)
ProTube for YouTube 2.0.2 Device: iOS Universal Category: Entertainment Price: $1.99, Version: 2.0.2 (iTunes) Description: ProTube is the ultimate, fully featured YouTube app. With it's highly polished design, ProTube offers ad-free... | Read more »

Price Scanner via MacPrices.net

15″ 2.2GHz Retina MacBook Pro on sale for $17...
 B&H Photo has the 2014 15″ 2.2GHz Retina MacBook Pro on sale today for $1749. Shipping is free, and B&H charges NY sales tax only. B&H will also include free copies of Parallels Desktop... Read more
27-inch 3.5GHz 5K iMac in stock today and on...
 B&H Photo has the new 27″ 3.5GHz 5K iMac in stock today and on sale for $2299 including free shipping plus NY sales tax only. Their price is $200 off MSRP, and it’s the lowest price available... Read more
21-inch 1.4GHz iMac on sale for $979, save $1...
B&H Photo has the new 21″ 1.4GHz iMac on sale for $979.99 including free shipping plus NY sales tax only. Their price is $120 off MSRP. B&H will also include free copies of Parallels Desktop... Read more
13-inch 1.4GHz/256GB MacBook Air on sale for...
B&H Photo has lowered their price on the 13″ 1.4GHz/256GB MacBook Air to $1059.99 including free shipping plus NY sales tax only. Their price is $140 off MSRP, and it’s the lowest price for this... Read more
Save up to $400 with Apple refurbished 2014 1...
The Apple Store has restocked Apple Certified Refurbished 2014 15″ Retina MacBook Pros for up to $400 off the cost of new models. An Apple one-year warranty is included with each model, and shipping... Read more
New 13-inch 1.4GHz MacBook Air on sale for $8...
 Adorama has the 2014 13″ 1.4GHz/128GB MacBook Air on sale for $899.99 including free shipping plus NY & NJ tax only. Their price is $100 off MSRP. B&H Photo has the 13″ 1.4GHz/128GB MacBook... Read more
Apple Expected to Reverse Nine-Month Tablet S...
Apple and Samsung combined accounted for 62 percent of the nearly 36 million branded tablets shipped in 3Q 2014, according to early vendor shipment share estimates from market intelligence firm ABI... Read more
Stratos: 30 Percent of US Smartphone Owners t...
Stratos, Inc., creator of the Bluetooth Connected Card Platform, has announced results from its 2014 Holiday Mobile Payments Survey. The consumer survey found that nearly one out of three (30 percent... Read more
2014 1.4GHz Mac mini on sale for $449, save $...
 B&H Photo has lowered their price on the new 1.4GHz Mac mini to $449.99 including free shipping plus NY tax only. Their price is $50 off MSRP, and it’s the lowest price available for this new... Read more
Check Apple prices on any device with the iTr...
MacPrices is proud to offer readers a free iOS app (iPhones, iPads, & iPod touch) and Android app (Google Play and Amazon App Store) called iTracx, which allows you to glance at today’s lowest... Read more

Jobs Board

*Apple* Solutions Consultant (ASC) - Apple (...
**Job Summary** The ASC is an Apple employee who serves as an Apple brand ambassador and influencer in a Reseller's store. The ASC's role is to grow Apple Read more
*Apple* Solutions Consultant (ASC)- Retail S...
**Job Summary** The ASC is an Apple employee who serves as an Apple brand ambassador and influencer in a Reseller's store. The ASC's role is to grow Apple Read more
Project Manager, *Apple* Financial Services...
**Job Summary** Apple Financial Services (AFS) offers consumers, businesses and educational institutions ways to finance Apple purchases. We work with national and Read more
*Apple* Store Leader Program - College Gradu...
Job Description: Job Summary As an Apple Store Leader Program agent, you can continue your education as you major in the art of leadership at the Apple Store. You'll Read more
*Apple* Retail - Multiple Positions (US) - A...
Sales Specialist - Retail Customer Service and Sales Transform Apple Store visitors into loyal Apple customers. When customers enter the store, you're also the Read more
All contents are Copyright 1984-2011 by Xplain Corporation. All rights reserved. Theme designed by Icreon.