Gamasutra - Features - "Exploring Spring Models" [10.05.01]
Contents
1. position xc current distance and xi distance at the inertial position Mathematically you can think of a spring as being two points separated from a distance x This is its inertial position In other words if you don t move these points no force will be generated because Dx 0 If you try to compress the distance between these points the Dx will be negative generating a positive force expansion If you try to separate these points the Dx will be positive generating a negative force compression The picture below illustrates these cases Pa I ININININVINVINININININININININ N Pb Dx 0 F 0 Pa N N IN V ININININININ N Pb Dx lt 0 F gt 0 Pa ININININININININININININININININININININ N Pb Dx gt 0 F lt 0 Springs using ascii art The k coefficient is called the elasticity coefficient and weights the spring s final force In other words a spring with bigger k is stiffer because it creates a larger force from its inertial state Conversely a spring with a smaller k is more flexible because it creates a smaller force from its inertial state 14 10 2004 00 23 Gamasutra Features Exploring Spring Models 10 05 01 http www gamasutra com features 20011005 oliveira_pfv htm Spring Force Implementation Implement the spring force is a direct application of the first two equations In other words first compute the distance of the spring s extremities Pa and Pb This i2. the connected springs This results into a realistic movement of connected points simulating what would happen in real life The three common applications of connected springs are string cloth and jelly These models are very similar The only difference is their initial geometric arrangement and how their Springs are connected For instance to model a string you connect springs as a line one dimension To model cloth you connect springs as a plane two dimensions Finally to model jelly you connect springs as a volume three dimensions Before implementing these applications take a closer look of how two springs can be connected Figure 1 shows two springs represented by two think colored lines with the extremities represented by three points The forces that can be created between these points are shown with arrows PO P1 P2 e 40 q4 Figure 1 Forces between Two Springs From Figurel two connected springs can generate four forces For example if you move and hold PO the left the force PO gt P1 will be created to pull PO back closer to P1 Also the force P1 gt P0 will be created to move P1 closer to PO As P1 is also connected to P2 the forces P1 gt P2 and P2 gt P1 will be also created to approximate these points If you had more points the forces will keep propagating through the springs because of its connections You don t have to necessarily follow the arrangement of Figure 1 For example Figure 2 shows four springs
3. vertices positions and links and setup the node s array one by one Fortunately there is a better ways to setup the cloth jelly and more complex spring models The main idea is to modify the creation s code for a more data driven procedure Data Driven Spring Models To be able to setup the springs in a data driven way suggests at first the need of a tool However you will probably have to spend a considerable amount of time and resources to implement this tool especially if you want to design complex 3d spring models Another way is to write a converter It turns out that by simply converting geometry data from a modeler tool 3D Max Maya Houdini Softimage etc to spring data you can still get good results The converter can save you a lot of time and has few advantages First you are not limited to symmetric geometry In other words the spring model is now only limited by your imagination Second you can get the texture information for your spring model for free more about this later Third as a videogame programmer you probably already have the geometry importer code available Therefore all you have to do is to modify it to create springs Fourth you do not have to worry about any fast algorithm during the conversion process Once you build your spring configuration simply save out the spring model Then just reload it later on the game The converter can be implemented by simply considering each vertex of the geometr
4. Gamasutra Features Exploring Spring Models 10 05 01 http www gamasutra com features 20011005 oliveira_pfv htm 1 of 18 jadooVWork i 9 YEARS OF NEW MAY 02 to APRIL 03 jacpot torka gaming WWW jadooworks com Gama Network Presents Gamasutra com Exploring Spring Models By Gustavo Oliveira Gamasutra October 5 2001 URL http www gamasutra com 20011005 oliveira_01 htm The first time implemented a spring model it fascinated me for hours The effect itself is amazingly realistic and its implementation is fairly simple Fortunately found a lot of articles references and source code to help with my research Nevertheless as went further down the road noticed that in most cases these references limited to the standard applications of spring models string cloth and jelly This article reviews the implementation of a spring model from its simplest form to more sophisticated applications taking the subject a step beyond the material available in most references Spring Basics Before modeling springs with the computer you need to understand the basic principles of Springs from your classic physics book As you compress or extend a spring it creates a force that is opposed to direction fo the force that you are applying This force is mathematically equated by the formula k Dx Dx XC X Where F is the resultant force k is the spring coefficient and Dx the distance from its inertial
5. R POSTE LON VECTOR vel current velocity VECTOR force resultant SPRING NODE neighbor_node SPRING_MAX NEIGHBOR_COUNT ft Loe t neighbor_x SPRING_MAX NEIGHBOR_COUNT distances INE neighbor_count SPRING _NODE The spring node data structure above holds the current position velocity resultant force and pointers to the neighbor nodes It also saves the distance x from the neighbor nodes at the inertial state necessary to compute the resultant force Yet Listing 2 is a little nasty Like Listing 1 it was written that way simply for educational purposes For instance this data structure is used for build time only Therefore to save some memory it would be better to use linked lists for the neighbor pointers and distances instead of arrays In a nutshell the spring model is simply a list array of spring nodes And to build a string or any spring model all you have to do is to set up the spring model initial geometric arrangement and connect its nodes properly Then you are ready to animate it by computing the internal forces and applying to the points Before you see the animation loop details take a closer look into the spring model data structure followed by the string creation code Listing 3 depicts the main data structures used to build any spring model As you see it is nothing more than a collection of nodes and extra fields for forces calculation and debugging Listing 4 is the actual setup code for a str
6. Tee 030 Ballon 2 sop _2 Energy Loss 099 Figure 18 Sample Program Screenshot k F Optimizations Cici on main windo Cancel and use ILI and M cerns ea ae Spring models rely heavily on square roots since the distance between two points is computed several times for each frame Luckily newer computers can handle these computations very easily for simple spring models Also the new consoles on the market have sets of assembler instructions to perform square roots and normalizations in single instructions by hardware For instance if you are a Playstation 2 developer Listing 1 can be easily optimized using VUO instructions At a higher level the way this article s code computes the forces is redundant If you look carefully at the code you are always computing the same force in two opposite directions Therefore it makes sense to compute the force in one direction and simply negate it to get its opposite By doing this way you are potentially cutting the inner calculations is almost half Sample Program The sample program demonstrates the standard applications string cloth and jelly anda more complex object balloon Play around with these models and see the spring s behavior in each object Use the interface buttons or keys I J K M Y and U move the objects Also the mouse press left or right button arrows key A and key Z move the camera The source code for the sample program is not available Only th
7. ass SPRING Listing 4 Creating a String SPRING SpringCreateString float distance int num_nodes float k_coef float energy_loss float mass SPRING Sprang if distance lt 0 005f num_nodes lt 3 return NULL eG LA EN ITALT AAN A VAI terre UAI LAAVAT AL IIVAA L LAAL I UIA UUA UAT allocate memory amp setup parameters TIPE TIT I AIA ALI LATAA LT IAT VL EL TALIA T LASLA IIA IAV TET AT Ine mem_size mem_size sizeof SPRING spring SPRING MemoryAllocAndClear mem_size allocate amp zero out memory TT CL Spring return NULLE adjust pointers spring gt k_coef k_coef spring gt energy_loss energy_loss spring gt mass mass LLALLA AATAL ATE III TTI IIAP TEI ATA III AT AP PTI SATA LUATA ILA LEVAN AAN setup nodes position aligned with the xz axis LILI LEAT ATI IATA EL IASI ILI CEA ETT TTA T AIT EIT ELT CPTI TI TIA TAA IELTS SPRING_NODE node_curr tr Emp VECTOR OOS Carr Um OTy Us Oe by Oils for tmpO 0 tmpO0 lt num_nodes tmp0 node_curr amp spring gt node tmp0 node_curr gt pos pos_curr pos CcUurrex distance WIE ELAAALAL I LIIS AID LAULA ET AAI LALIT PIAA III EP TIA FTAA PLATE ITT LAA STAT TALITY connect nodes and setup the inertial x distances TTI TPL APT IAA TI VITA ATTA ITI a IAA AAPL IAAT TET II tj tar left mode node_curr amp spring gt node 0 node_curr gt neighbor_node 0 amp spring gt
8. below shows the implementation it s a little redundant for sake of clarity null out the force in case of potential division by zero Listing 1 Computing A Spring s Force define SPRING_TOLERANCE 0 0000000005F void SpringComputesinglerForce VECTOR force VECTOR pa VECTOR pb float k float delta distance VECTOR Forme e drr float intensity float distance float delta get the distance float dx PO gt x pax PLO Gy Poey pan y float dz Pos pa 2j dictante S I loat Sgr dx dx tay ady dz dz if distance lt SPRING _ TOLERANCE force gt x force gt y force gt z 0 0f return force dir dx force direy dy torce dir z dz normalize force _dir x distance force_dir y distance force diriz distance force delta distance delta_distance intensity k delta store 2 of 18 14 10 2004 00 23 Gamasutra Features Exploring Spring Models 10 05 01 http www gamasutra com features 200 1 1005 oliveira_pfv htm 3 of 18 force gt x force_dir x intensity force gt y force_dir y intensity force gt z force_dir z intensity Connecting Springs Computing the spring force in one direction for a single spring doesn t do much The power of Springs comes when you connect them into a network Then the computed force of one spring affects the entire network In other words if one point moves it creates a force that propagates through
9. connected together in a different manner Again the thick colored lines represent the springs the points its extremities and the arrows the forces that can be created between them P3 PO P4 Figure 2 Forces between Four Springs 14 10 2004 00 23 Gamasutra Features Exploring Spring Models 10 05 01 http www gamasutra com features 2001 1005 oliveira_pfv htm 4 of 18 In this arrangement there are four possible acting forces on P1 derived from PO P2 P3 and P4 Also you don t necessarily have to connect a point to its neighbors as in the previous figures For instance you can add another spring connecting PO to P2 Or even add more than one spring connecting two points In fact they way you build your spring network is entirely up to you The arrangement you choose however will affect your final spring model behavior Based on this idea the figures below show ways to connect springs for a string a cloth and a jelly respectively In each arrangement the edges represents one spring The blue dots are the extremities of the springs Figures 3 5 String Arrangement Cloth Arrangement J elly Arrangement In each configuration if you move one point it will realistically simulate the object s behavior However this is not entirely true for the particular jelly arrangement presented Indeed the arrangement of Figure 5 is unstable If you model a jelly exactly like in Figure 5 moving one of its points wil
10. e acceleration value you will get fine results This is equivalent of having set the mass of all springs equals to one In the code you see the line with the extra division commented out Another important observation is that the code adds gravity to the final resultant force implemented by the global variable gSpringGravity This is simply an additional global force that affects all nodes making the string fall You can take this out or even add more of these global forces for different results The last important observation is in regard to the calculation of the node s velocity The acceleration value adds velocity to the node but you need to dampen this velocity somehow each frame If you don t your spring model will swing forever The energy loss at the end of Listing 7 is used to dump the node s current velocity It is interesting to play around with the energy_loss variable For example if you set it to 1 0 you will get perpetual motion If you set this variable slightly above 1 0 on every frame the system will gain energy Eventually your spring model will be completely chaotic and unrecognizable Realistic values would be between 0 0 0 percent to 1 0 100 percent but non inclusive This will simulate real life energy loss of moving systems due to attrition heat and or noise When first saw my string swinging beautifully on my screen had not included my 9 of 18 14 10 2004 00 23 Gamasutra Features Exploring Sp
11. e string implementation is available but with no render functions and no optimizations The sample code is just to guide you on your implementation Final Comments The more you play with springs the more ideas you have for things you can model with them Objects modeled with springs can add more realism to your game Yet there are not many games out exploring this neat effect 14 10 2004 00 23 Gamasutra Features Exploring Spring Models 10 05 01 http www gamasutra com features 200 1 1005 oliveira_pfv htm 18 of 18 Next time you would like to write an interesting effect for your game see if springs can give you new ideas Like the famous Duke Ellington s standard says It Don t Mean a Thing If It Ain t Got That Swing References 1 http freespace virgin net hugo elias 2 Serway Physics For Scientists and Enginneers Fourth Edition pg 364 370 3 Watt Alan and Fabio Policarpo The Computer Image Addison Wesley 1997 4 Harris John W and Horst Stocker Handbook of Mathematics and Computational Science Springer Verlag 1998 5 Donald Hearn M Pauline Baker Computer Graphics C Version 2nd ed 6 Direct X 8 SDK Help Microsoft Corporation 7 Playstation 2 VU User s Manual 2nd Edition Sony Computer Entertainment Inc FEditing Copyright 2003 CMP Media Inc All rights reserved 14 10 2004 00 23
12. e string nodes Listing 9 shows the render function that does that and also draws Small crosses at each node for reference Figures 9 10 and 11 and 12 shows screen shots of the string movement with some gravity value Listing 9 Rendering The String RESULT SpringRenderString SPRING spring a bea spring return SPRING _ERROR_PARAM IS_NULL SPRING _NODE node_a SPRING_NODE node_b int tmpt RGBA rgba for tmpo 0 tmp0 lt spring snode counc 1l tmpo0 node_a amp spring gt node tmp0 node _ b amp spring gt node tmp0 1 SET RGBA amp rgba 0x80 0x40 Oxa0O Oxff LineRender amp node_a gt pos amp node_b gt pos amp rgba SET_RGBA amp rgba Oxf0O OxtO Oxt0 0x00 draw little crosses LineRenderCross amp node_a gt pos 1 0f amp rgba LineRenderCross amp node_b gt pos 1 0f amp rgba return SPRING_OK 14 10 2004 00 23 Gamasutra Features Exploring Spring Models 10 05 01 http www gamasutra com features 20011005 oliveira_pfv htm Figures 9 12 Animation of the string Implementing Cloth and Jelly 12 of 18 14 10 2004 00 23 Gamasutra Features Exploring Spring Models 10 05 01 http www gamasutra com features 200 1 1005 oliveira_pfv htm 13 of 18 If you understood the basic idea of the string s algorithm implementing cloth and jelly is literally the exact same thing You can simply use the same code yo
13. e_curr gt vel z spring gt energy_loss Once the string starts to swing the last part of the code just checks to see if the anchor point has moved from its initial position If is has the code forces it back to the original location In the case of the string the anchor point was set be one the strings extremities This works like you had nailed the string s extremity to a wall one end will never move If you hook up the keyboard or mouse to the anchor point position you can move the string around to see how it behaves Listing 8 illustrates this simple check void SpringAnchorCheck Listing 8 Checking Anchor Points iene tmpO SPRING NODE node curr SPRING springm test new position against the anchor for tmp0 0 tmp0 lt soring sanchor count tmpUa node_curr spring gt anchor tmp0 node 14 10 2004 00 23 Gamasutra Features Exploring Spring Models 10 05 01 11 of 18 if mode _ curr gt pos x spring gt anchor tmp0 pos x node_curr gt pos x spring gt anchor tmp0O pos x it node_curr gt pos y spring gt anchor tmp0 pos y i Node Curr pos 7 spring gt anchor tme0 pos y if node_curr gt pos z spring gt anchor tmp0 pos z node curr gt p0s 7 spring gt anchor tmp0 pos z http www gamasutra com features 2001 1005 oliveira_pfv htm Lastly to draw the string simply hook up a render function that draws lines using the current position of th
14. ing It has four main parts First it allocates memory for the string data and assigns the main parameters to it Second it positions the nodes aligned to the xz positive axis Third connects the nodes and assigns the inertial distances for the nodes and its neighbors The last part is where the spring model behavior is defined This part of Listing 4 is fairly simple and hopefully easy to read That s because it models a string and the implementation takes advantage of the string s symmetry The last part of Listing 4 is setting up the anchor point This is a point that will not animate no matter how strong the forces are on it The anchor points are used for the debugging and testing To drag model around you need at least point In some models as you will see later the anchor points are useful to keep the spring model stable Listing 3 Spring Model Data Structure define SPRING MAX NODE COUNT 4096 define SPRING_MAX_ANCHOR_NODE_COUNT 236 typedef struct spring anchor VECTOR DOs SPRING NODE node OPRING_ANCHOR typedef struct _spring SPRING NODE node SPRING_MAX NODE COUNT array of spring nodes coke node Counc 6 of 18 14 10 2004 00 23 Gamasutra Features Exploring Spring Models 10 05 01 http www gamasutra com features 20011005 oliveira_pfv htm 7 of 18 SPRING_ANCHOR anchor SPRING_MAX ANCHOR_NODE COUNT stability amp debug Ti anchor count float k_coef ElLoat energy_loss float m
15. l morph the jelly into a different object Soon the original cubical shape will be be gone Why Because the spring force formula requires the scalar Dx and its direction is irrelevant Therefore if you simply rotate two connected springs they can move to another configuration where no forces will be created Figure 6 and Figure 7 below clarify the last statement by showing two sets of springs that do not create any forces In both cases the Dx 0 but their geometric shape are different 14 10 2004 00 23 Gamasutra Features Exploring Spring Models 10 05 01 http www gamasutra com features 20011005 oliveira_pfv htm 5 of 18 PO Pi P2 amp E CO PO P1 Figures 6 7 Inertial Springs Dx 0 Inertial Springs Dx 0 Because the springs can rotate around their extremities without creating any forces you must be careful when modeling and connecting your springs This is probably the most important part of the implementation You must design your spring model to be able to swing nicely without morphing it into something unrecognizable Also your design should use the least number of springs for the sake of calculation time Figure 8 composes a stable arrangement for a jelly model In this arrangement there are a lot more springs than the previous one and some vertices connect up to six springs This is still not the only way of connecting them to create a stable model For example you can stack the cloth model and connect all the
16. m and top parts of the jelly To fine tune your springs you can write a simple tool that let you attach or delete springs from your current spring model This tool is far less complicated than writing a spring modeler from scratch Figure 13 shows the tool used to convert and edit the springs of the jelly in Figure 12 to the one in Figure 8 K Coeicent 0130 I wire Frame Ereg Loss 0195 Texture Braet 00 03 00 Connections amie B _ dfo 05 00 00 Maks Figure 13 Conversion And Editing Spring Tool mr The conversion tool in figure 13 is simple mesh viewer with few extra buttons for spring editing The interface allows the user loop through the spring nodes add or delete springs add anchor nodes test the model and tune the physics parameters Despite this tool interface s Simplicity it allows the user to convert tune and test the spring model in a data driven way Texturing and Lighting Spring Models So far all you have seen in this article are wireframe examples but to texture them Is trivial When converting the geometry data to a spring model the texture information comes with no effort Simply move it from your original mesh data to the spring data structure Also you need to move the face data structure In other words to draw each triangle you need to know which nodes compose each face Listing 10 illustrates the basic modifications to the main data structure to include texture informatio
17. n 14 10 2004 00 23 Gamasutra Features Exploring Spring Models 10 05 01 http www gamasutra com features 2001 1005 oliveira_pfv htm Listing 10 Spring Model Data Structure With Texture I nformation typedef struct _spring_face int 10 int Il int LA vertex index int tO int El nE E2 texture index SPRING_FACE typedef struct _spring_tvert float us float Vv SPRING TVERT typedef struct _spring SPRINGM_TVERT SPRING_NODE int node count SPRING_ANCHOR debug In anchor Count SPRING _FACEK face disi int face _list_count SPRING_TVERT tvert List int tvert List count float k coef float energy_loss float mass SPRING node SPRING_MAX NODE COUNT array of spring nodes anchor SPRING_MAX ANCHOR_NODE_COUNT stability amp array of face indices array of texture coordinates Figures 14 15 16 and 17 show a textured cloth model and four animation frames The cloth was build by the conversion tool Note that the red edges were deleted springs since the original geometry does not create a good spring structure for a cloth Four anchor points were set in a Shape of a square around the center of the cloth 15 of 18 14 10 2004 00 23 Gamasutra Features Exploring Spring Models 10 05 01 http www gamasutra com features 20011005 oliveira_pfv htm 16 of 18 Figures 14 17 Textured Cloth Dynamic lighting is more complicated since your spri
18. ng model is constantly morphing In this case you must compute the vertex normals on the fly for each frame and there is no way around it The standard way is to setup pointers to neighbor faces in build time Then in real time compute and average the neighbor face normals to get each vertex normal Complex Spring Models With a data driven spring system you can pretty much model and design anything you want to swing You can build balloons trees tires water hair tents and so on Even the standard applications can be enhanced like a flag with a flexible pole for example Another interesting effect is to mix strings with cloth or jelly like objects You can create a cool balloon by connecting a string to a spherical jelly Another example would be for a F1 game modeling the finish line cloth with a cloth connected by strings to the poles As the cars cross the line apply the forces due to the air movement to the spring model Figure 18 shows the screen shot of this article s executable There you will find few spring models It has a string a cloth a jelly and a balloon Except for the string they were modeled with the conversion tool 14 10 2004 00 23 Gamasutra Features Exploring Spring Models 10 05 01 http www gamasutra com features 20011005 oliveira_pfv htm 17 of 18 pring Demo Dustavo s Soring Flay oumd Selection Movemert _ Pasatnatinrs H 5 z o Grevay 0 0 0 03 0 0 Bod et a
19. node 1 node_curr gt neighbor_x 0 distance node_curr gt neighbor_counttt ji Lat rignt node node_curr amp spring gt node num_nodes 1 node_curr gt neighbor_node 0 amp spring gt node num_nodes 2 node_curr gt neighbor_x 0 distance 14 10 2004 00 23 Gamasutra Features Exploring Spring Models 10 05 01 http www gamasutra com features 20011005 oliveira_pfv htm node_curr gt neighbor_countt t inner nodes for tmp0 1 tmpO lt num_nodes 1 tmp0 node_curr amp spring gt node tmp0 Connect to the previous node_curr gt neighbor_node 0 amp spring gt node tmp0 1 node_curr gt neighbor_x 0O distance connect to the next node_curr gt neighbor_node 1 amp sSpring gt node tmp0 1 node_curr gt neighbor_x 1 distance node curr neighbor count 2 PI PATI AA ALITA SIA ALAA AAI AAT EAT ST setup anchor node IAAL IAL EI TAIALA LEA LUI TA PT ALLE Fitar right node isthe only anchor spring gt zanchor Ol Hoda S spring snode 0 spring gt anchor 0 pos spring gt node 0 pos return spring Once the geometry is ready and the nodes are connected the next step is to implement the animation loop Listing 5 illustrates the generic loop Again this code is redundant and not optimized for sake of clarity The basic idea is to loop through all the nodes compute all the forces on each node and store the resultant force Then the resulta
20. nt is used to add velocity to the node s position Finally the anchor points are fixed by not letting the forces act on them Listing 5 Animation Loop void SpringComputeForces SPRING spring SpringAccumulateForces Spring SpringApplyForces spring SpringAnchorCheck spring To compute the final resultant force on each node the neighbor nodes come into play In Listing 6 for each node you compute the forces that all neighbor nodes apply to it Then you accumulate add all these forces and store it in the data structure Do you remember in Listing 1 when only one direction was necessary to compute the force This is because as you are looping through all nodes and looping through all its neighbors as well Therefore at the end you will be computing both directions of the same force In other words when PO is the current node the force P1 gt P0 is computed Then when the current node is P1 the force PO gt P1 is computed As you are computing the same force with opposite signs twice this can be optimized later As of right now to keep the loop easier to read the code does it the Slow way Listing 6 Accumulating the Forces on Each Spring Node void SpringAccumulateForces SPRING spring INE tmel int Emo0 VEC TOR rorce tmp SPRING NODE node curr WiC LOR source VECTOR dest float distance 8 of 18 14 10 2004 00 23 Gamasutra Features Exploring Spring Models 10 05 01 http www gamasut
21. planes with more springs This configuration will be computationally more intense and it will behave differently than the arrangement of Figure 8 Again the final arrangement is up to you and it depends on what you are trying to simulate Figure 8 Stable Arrangement For Jelly Implementing a String With the basic information already presented let s look in how to implement the simplest Spring model a string Despite of its simplicity once you understand how to implement a string other models become just an extension to the same basic idea To begin you need a data structure that allows you to save the spring s position compute its forces and enable connection to other springs Therefore it is easier to design your data for the spring s extremities instead of each spring itself In this implementation each extremity of a spring is called a spring node Two spring nodes compose one spring As a polygonal mesh has a list of vertices the spring model will have a list of spring nodes The code below shows the basic data structure for each spring node and the helper data structure VECTOR Listing 2 Spring Node Data Structure 14 10 2004 00 23 Gamasutra Features Exploring Spring Models 10 05 01 http www gamasutra com features 2001 1005 oliveira_pfv htm define SPRING_MAX NEIGHBOR_COUNT 128 typedef struct _vector f Loat xs te Loa Vi float Zz VECTOR typedef struct _spring_node VECTOR pos COC
22. ra com features 2001 1005 oliveira_pfv htm VECTOR resultant faccumulate all forces for Empl 0s tmp lL lt spring node count tmpol node_curr amp spring gt node tmpl Anali Gut resultant res ltant x resultant y resultant z 0 0f source amp node_curr gt pos for tmp0 0 tmp0 lt node_curr gt neighbor_count tmp0O dest amp node_curr gt neighbor_node tmp0 gt pos distance node_curr gt neighbor_x tmp0 SpringComputeSingleForce amp force_tmp source dest spring gt k_ coef distance VectorAdd amp resultant amp resultant amp force_tmp node_curr gt force resultant To finally animate the spring model you simply apply the resultant force of each to node to itself making them to move From your physics book force and acceleration are related as following F M Acc Acc F M As the first equation shows to compute the node s acceleration you divide the resultant force by the spring s mass If the force is not null there will be an acceleration value that is used to increase the velocity of the node Finally the velocity will change the node s position each frame Listing 7 shows the implementation of all that There are few important observations about Listing 7 To begin if you remove the spring s mass value from the code it will not make much of a difference Although to do the extra divide is mathematically correct but if you simply assign the force to th
23. ring Models 10 05 01 10 of 18 http www gamasutra com features 2001 1005 oliveira_pfv htm energy_loss variable to the code For a second truly thought had invented a way to create perpetual motion However had forgotten that my computer was plugged to the outlet Therefore keeping my pixels moving forever would cost me infinite energy Listing 7 Applying Forces To The Nodes void SpringApplyForces SPRING spring TURER EME SPRING_NODE node curr VECTOR acc apply resultant forces to increase vel of each spring for tump0 0 gt tmp0 lt spring Snode count node_curr amp spring gt node tmp0 mass iS in somehow unecessary EmpO fF fACCER node curr gt force x gSpringGravity x spring gt mass acc xX node curr gt force x tgSpringGravity x J fACCAY node_curr gt force ytgSpringGravity y spring gt mass ACC ay node_curr gt force ytgSpringGravity y LT ACE cZ node CGurr sTOrGe ZIOSpringGravivty lt 7 Spring gt mass ACC Z node curr gt Lorce ZztgSspringGravity lt z node curr val node Curr vel node_curr gt pos node_curr gt pos cx Y node_curr gt vel z x Y node_curr gt pos z acc x t aCC V3 aAcc Z f node curr gt vel node curr gt vel node_curr gt vel ffenergy Voss Tor velocity node_curr gt vel x spring gt energy_ xy Va Zi loss node_curr gt vel y spring gt energy_loss nod
24. s done through Pythagoras theorem Once you have the distance you subtract from the inertial distance getting the Dx Then multiply this scalar value to the k coefficient and finally use the inverse of this value to compute the force As simple as it sounds the implementation in C code has its details First the distance Dx and the k coefficient are scalars but the force itself is a vector Therefore it requires a direction which can be from Pa to Pb or from Pb to Pb As of right now the code below always use the direction Pb to Pa or mathematically Fdir Pb Pa As the force s direction is normalized you can use the distance you already computed for the Dx for the normalization You must be careful is to avoid division by zero when normalizing the vector This is a nasty condition and it usually does not happen but you have to do something about it in case it does You have two options in this particular case assign a random force to the points or null out the force Theoretically the force should be big enough that as the points get closer they will never occupy the same space Therefore it seems to be a good option to randomly assign a massive force in this case However this can cause undesirable artifacts because some points inside the spring model might fly a part with extreme velocity when compared with others Although nulling out seems wrong it doesn t cause any major problems and preserves the spring s stability The code
25. u have written to simulate a string for more complex objects The only part you need to change is the creation function Listing 4 Then you will have to arrange the position of the nodes and the links to its neighbors to fit the cloth and jelly shapes As you may have notice Listing 4 was fairly simple because a string is a simple object to setup The cloth is not too bad to setup because it is also a very symmetric object Yet compared to the string the code will be a lot more complicated For the cloth you are going to have to setup the nodes position s as a square shape Then when connecting the nodes you need to be careful with the corners left right top and bottom edges and the inner parts of the cloth And how about the jelly It is possible algorithmically to setup the jelly like a stack of cloth Yet the same setup code is now even more complicated than the cloth s code First you need to setup the node s initial position as a cubic shape Then when connecting the nodes you need to be careful with the corners front back left right top and bottom planes and the inner parts of the jelly AS you can see in this case you may have to spend a lot more time on the jelly s creation code than the physics itself To make matters worse if you try to model algorithmically a jelly like the one in Figure 8 the code is going to get even more tangled up In this particular case you are better offer to get a graphic paper and write out the
26. y vertex list a Spring node Then each edge connecting two vertices becomes a spring The basic algorithm is shown next for all vertices get Current vertex for all faces seek neighbor vertices to the current vertex save the neighbor vertex and the x distance to them loop faces loop vertices Although writing a converter offers several advantages it does have a big disadvantage Your spring model now is hooked to the geometry data and this is not necessarily a good way to setup your springs Most of 3D tools export geometry as a list of triangles Therefore your Springs will always have its inner cells connected as triangles Figure 12 depicts the jelly model arrangement converted from geometry data 14 10 2004 00 23 Gamasutra Features Exploring Spring Models 10 05 01 http www gamasutra com features 20011005 oliveira_pfv htm 14 of 18 Figure 12 Converted J elly Arrangement When perturbing and applying the spring forces to this jelly arrangement it will morph into something unrecognizable In other words this arrangement is unstable Now if writing a tool is too much work but converted geometry data does not work what can you do In a nutshell you can do the gross arrangement of the spring model through the converter and manually add or remove springs to fine tune your model If you compare Figure 12 with Figure 8 you will realize that they are similar The only difference is few inner springs connecting the botto
Download Pdf Manuals
Related Search