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!

 

Community Search:
MacTech Search:

Software Updates via MacUpdate

Dash 4.0.3 - Instant search and offline...
Dash is an API documentation browser and code snippet manager. Dash helps you store snippets of code, as well as instantly search and browse documentation for almost any API you might use (for a full... Read more
Posterino 3.3.6 - Create posters, collag...
Posterino offers enhanced customization and flexibility including a variety of new, stylish templates featuring grids of identical or odd-sized image boxes. You can customize the size and shape of... Read more
Apple Numbers 4.1.1 - Apple's sprea...
With Apple Numbers, sophisticated spreadsheets are just the start. The whole sheet is your canvas. Just add dramatic interactive charts, tables, and images that paint a revealing picture of your data... Read more
Apple Pages 6.1.1 - Apple's word pr...
Apple Pages is a powerful word processor that gives you everything you need to create documents that look beautiful. And read beautifully. It lets you work seamlessly between Mac and iOS devices, and... Read more
iClock Pro 3.4.9 - Customize your menuba...
iClock Pro is a menu bar replacement clock for Apple's default clock. iClock Pro is an update, total rewrite and improvement to the popular iClock. Have the day, date and time in different fonts and... Read more
Typinator 7.2 - Speedy and reliable text...
Typinator turbo-charges your typing productivity. Type a little. Typinator does the rest. We've all faced projects that require repetitive typing tasks. With Typinator, you can store commonly used... Read more
ExpanDrive 5.4.6 - $49.99
ExpanDrive builds cloud storage in every application, acts just like a USB drive plugged into your Mac. With ExpanDrive, you can securely access any remote file server directly from the Finder or... Read more
Espresso 5 - Powerful HTML, XML, CSS, an...
Espresso is back. For people who make delightful, innovative, and fast websites -- in an app to match. Espresso helps you write, code, design, build, and publish with flair and efficiency.... Read more
Things 2.8.12 - Elegant personal task ma...
Things is a task management solution that helps to organize your tasks in an elegant and intuitive way. Things combines powerful features with simplicity through the use of tags and its intelligent... Read more
Apple Keynote 7.1.1 - Apple's prese...
Easily create gorgeous presentations with the all-new Keynote, featuring powerful yet easy-to-use tools and dazzling effects that will make you a very hard act to follow. The Theme Chooser lets you... Read more

Latest Forum Discussions

See All

Minimalist puzzle game, Cuts, free on iO...
If you're looking for a gorgeous puzzle experience on iOS devices, developer Gamebra.in's aesthetically interesting puzzler, Cuts, is discounted to free on the iOS App Store right now. [Read more] | Read more »
Anime tactical RPG, War of Crown, comes...
If you're looking for another tactical RPG fix to go alongside your Fire Emblem Heroes campaigns check out Gamevil's newest, anime-inspired tactics RPG, War of Crown, which comes out tomorrow. [Read more] | Read more »
Fantasy MMORPG MU Origin adds new modes,...
MU Origin, Webzen’s highly popular fantasy MMORPG is getting ready to shake things up for the second time this year, as a new update makes its way to the Google Play and App Store from today. Introducing new systems, modes, and events, the land of... | Read more »
Blizzard is looking to hire a mobile dev...
A new thread on the popular video game rumor forum, NeoGAF, uncovered an interesting job listing over at Blizzard Entertainment. It appears the studio behindStarCraft, World of WarCraft, Hearthstone,andOverwatch is looking to bring on a new hire... | Read more »
Legend of Zelda meets Cooking Mama in ne...
Dungeon Chef is what happens when you mix the RPG elements (and style) of a Legend of Zelda game, with cooking elements. Although, now that The Legend of Zelda: Breath of the Wild also has cookingelements, so maybe the gameplay is not so novel.... | Read more »
ChordFlow (Music)
ChordFlow 1.0.0 Device: iOS Universal Category: Music Price: $6.99, Version: 1.0.0 (iTunes) Description: ChordFlow is a chord sequencer with a unique 4-track polyphonic arpeggiator, extensive chord library, MIDI out and Ableton Link... | Read more »
The Walking Dead: A New Frontier is out...
The newest season of Telltale Games'The Walking Dead is well underway. After the release of the third episode, "Above the Law" about a month ago, episode four, "Thicker Than Water" is hot and ready for more zombies and gut-wrenching emotional... | Read more »
Best games we played this week
Another week, another new wave of mobile games do dive into. We've dug through the list of apps that came out this week to tell you which apps are worth your sweet time. And while there weren't too many games this week, there were some big ones.... | Read more »
Vignettes (Games)
Vignettes 1.0.1 Device: iOS Universal Category: Games Price: $2.99, Version: 1.0.1 (iTunes) Description: Vignettes is a casual but unique exploration game without text or characters, where objects shapeshift as you spin them around... | Read more »
Get Me Outta Here is an 80s retro shoote...
Are you ready to fight some aliens? Because Crescent Moon Games has released the retro shooter Get Me Outta Here on iOS devices today. [Read more] | Read more »

Price Scanner via MacPrices.net

HyperX Ships Pulsefire FPS Gaming Mouse, Winn...
Your reporter is a longtime fan of gaming mice for general purpose coomnputing use, finding them typically superior in comfort and performance. HyperX, a division of Kingston Technology Company, Inc... Read more
Penske Truck Leasing Unveils “Penske Fleet” M...
Penske Truck Leasing has introduced a free mobile app called “Penske Fleet” to benefit its full-service truck leasing and contract maintenance customers. The mobile app enables Penske’s customers to... Read more
DEVONthink To Go 2.1.3 For iOS Adds New Declu...
DEVONtechnologies has updated DEVONthink To Go, its iOS companion to DEVONthink for Mac. Enhancements and bug fixes in Version 2.1.3 include DEVONtechnologies’ new decluttering service for clipping... Read more
Use Apple’s Education discount to save up to...
Purchase a new Mac or iPad using Apple’s Education Store and take up to $300 off MSRP. All teachers, students, and staff of any educational institution qualify for the discount. Shipping is free: -... Read more
Apple refurbished Apple Watches available sta...
Apple is now offering Certified Refurbished Series 1 and Series 2 Apple Watches for 14-16% off MSRP, starting at $229. An Apple one-year warranty is included with each watch. Shipping is free: Series... Read more
15-inch Touch Bar MacBook Pros, Apple refurbi...
Apple is offering Certified Refurbished 2016 15″ Touch Bar MacBook Pros for $360 to $420 off original MSRP. An Apple one-year warranty is included with each model, and shipping is free: - 15″ 2.6GHz... Read more
13-inch MacBook Airs on sale for up to $150 o...
Overstock.com has 13″ MacBook Airs on sale for up to $150 off MSRP including free shipping: - 13″ 1.6GHz/128GB MacBook Air (sku MMGF2LL/A): $869.99 $130 off MSRP - 13″ 1.6GHz/256GB MacBook Air (sku... Read more
15-inch Touch Bar MacBook Pros on sale for $1...
B&H Photo has the new 2016 15″ Apple Touch Bar MacBook Pros in stock today and on sale for up to $200 off MSRP. Shipping is free, and B&H charges NY sales tax only: - 15″ 2.7GHz Touch Bar... Read more
15-inch 2.7GHz Touch Bar MacBook Pros on sale...
Amazon has 2016 15″ 2.7GHz Apple Touch Bar MacBook Pros in stock today and on sale for $150-$200 off MSRP. Shipping is free: - 15″ 2.7GHz Touch Bar MacBook Pro Space Gray (sku MLH42LL/A): $2599 $200... Read more
Apple now offering Certified Refurbished 13-i...
Apple is now offering Certified Refurbished 2016 13″ Touch Bar MacBook Pros for $270-$300 off original MSRP. An Apple one-year warranty is included with each model, and shipping is free: - 13″ 2.9GHz... Read more

Jobs Board

*Apple* Engineering Specialist - CSRA (Unite...
Apple Engineering Specialist All times are in Eastern Daylight Time Requisition ID Job Locations US DC Washington DC Posted Date Category Engineering Sciences Read more
*Apple* Mac Computer Technician - GeekHampto...
…complex computer issues over the phone and in person? GeekHampton, Long Island's Apple Premium Service Provider, is looking for you! Come work with our crew Read more
*Apple* Retail - Multiple Positions - Apple,...
Job Description: Sales Specialist - Retail Customer Service and Sales Transform Apple Store visitors into loyal Apple customers. When customers enter the store, Read more
*Apple* Solutions Consultant - Apple (United...
# Apple Solutions Consultant Job Number: 56881986 Middletown, NY, New York, United States Posted: Apr. 17, 2017 Weekly Hours: 40.00 **Job Summary** As an Apple Read more
*Apple* Mobile Master - Best Buy (United Sta...
**489879BR** **Job Title:** Apple Mobile Master **Location Number:** 000402-Utica-Store **Job Description:** **What does a Best Buy Apple Mobile Master do?** At Read more
All contents are Copyright 1984-2011 by Xplain Corporation. All rights reserved. Theme designed by Icreon.