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  • How to tile multiple procedurally generated textures?

    - by Burhuc
    I'm trying to develop a procedural tile generator for a game, mostly for the ground tiles, instead of using "hand-drawn" tiles. To achieve this I'm using Perlin noise and a sine wave with multiple parameters, which already gives me pretty nice results. I don't want to generate 1 tile and repeat that one forever for one ground type, but I want to avoid recurrences, so I'm generating n different tiles. The problem I'm having now is that I want to tile the generated textures (smooth transitions). At the moment I have this: 4 256x256 textures. I thought a simple method would be to just add the positions of the different tiles to the noise generation algorithm, so that, when creating the 4 256x256 textures, it would behave like it would create a 512x512 texture, but that somehow didn't work as intented. So how can I tile those textures?

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  • Box2D `ApplyLinearImpulse` is not working whereas `SetLinearVelocity` works

    - by Narek
    I need to mimic jumping behavior for the player in my game. Player consists of two fixtures with circle and rectangle shapes. Rectangle I use to detect ground and it is a sensor. Is some point for jumping I do this: float impulseY = body->GetMass() * PLAYER_JUMPING_VEOCITY / PTM_RATIO * std::sin(PLAYER_JUMPING_ANGLE * PI / 180); body->ApplyLinearImpulse(b2Vec2(0, impulseY), body->GetWorldCenter(), true); and player does not jump. But when I do this: body->SetLinearVelocity(b2Vec2(0, PLAYER_JUMPING_VEOCITY / PTM_RATIO * std::sin(PLAYER_JUMPING_ANGLE * PI / 180))); my player jumps. Also when I change the rectangle shape to be normal (not sensor) shape, its works again. Why? Just in case here are the parameters of my rectangular sensor: b2PolygonShape boxShape; boxShape.SetAsBox(width * 0.5/2/PTM_RATIO, height * 0.2/2/PTM_RATIO, b2Vec2(0, -height * 0.4 /PTM_RATIO), 0); b2FixtureDef boxFixtureDef; boxFixtureDef.friction = 0; boxFixtureDef.restitution = 0; boxFixtureDef.density = 1; boxFixtureDef.isSensor = true; boxFixtureDef.userData = static_cast<void*>(PLAYER_GROUP);

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  • Bouncing ball isssue

    - by user
    I am currently working on the 2D Bouncing ball physics that bounces the ball up and down. The physics behaviour works fine but at the end the velocity keep +3 then 0 non-stop even the ball has stopped bouncing. How should I modify the code to fix this issue? ballPos = D3DXVECTOR2( 50, 100 ); velocity = 0; accelaration = 3.0f; isBallUp = false; void GameClass::Update() { velocity += accelaration; ballPos.y += velocity; if ( ballPos.y >= 590 ) isBallUp = true; else isBallUp = false; if ( isBallUp ) { ballPos.y = 590; velocity *= -1; } // Graphics Rendering m_Graphics.BeginFrame(); ComposeFrame(); m_Graphics.EndFrame(); }

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  • Why does Unity in 2d mode employ scaling and the default othographic size the way it does?

    - by Neophyte
    I previously used SFML, XNA, Monogame, etc to create 2d games, where if I display a 100px sprite on the screen, it will take up 100px. If I use 128px tiles to create a background, the first tile will be at (0,0) while the second will be at (129,0). Unity on the other hand, has its own odd unit system, scaling on all transforms, pixel-to-units, othographic size, etc etc. So my question is two-fold, namely: Why does Unity have this system by default for 2d? Is it for mobile dev? Is there a big benefit I'm not seeing? How can I setup my environment, so that if I have a 128x128 sprite in Photoshop, it displays as a 128x128 sprite in Unity when I run my game? Note that I am targeting desktop exclusively.

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  • Memory allocation strategy for the vertex buffers (DirectX 10/11)

    - by Alex
    I have the following question. I write CAD system. So I have a 3D scene and there are many different objects (walls, doors, windows and so on). User can add or delete some objects. The question is: how can I organise the keeping of vertices for all my objects. I can create vertex buffer for every object. But I think drawing/switching from one buffer to another would have performance penalty. Another way - I can create several big buffers for every object type. But I don't understand how to update such buffers. It is too big to update whole buffer (for example buffer for all walls). What I need to do if I want to delete the object from the middle of the buffer? Actually I have the similar question: http://stackoverflow.com/questions/5515700/how-to-properly-update-vertex-buffers-in-directx-10 Most examples I've found work with very static models. Therefore, they tend to create a single vertex buffer with their list of points, and then are just manipulated by matrix transformations. I, on the other hand, will be updating the scene very often.

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  • Has a multi player graphic adventure* ever been made?

    - by Petruza
    By graphic adventure, I mean point & click LucasArts-type games. Those games have a mostly linear structure in nature, and usually don't offer as many variants as other games types like action, rpg, strategy, which makes this genre difficult to implement a multi-player feature. I'd like to know if there has been any attempts on doing such a thing, and if it would be viable, as players going offline or leaving a game in the middle would affect significantly the other players' game.

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  • Find angle for projectile to meet target in parabolic arc

    - by TheBroodian
    I'm making a thing that launches projectiles in 2D. Its projectiles are fired with a set initial velocity, and are only affected by gravity. Assuming that its target is within range, and that there aren't any obstacles, how would my thing find the appropriate angle at which to launch its projectile (in radians)? The equation for this is found here: Wikipedia: Angle Required to Hit Coordinate Sadly, I'm not a physicist (a.k.a. can't read smart people math) and am having a hard time reading its breakdown. If not only for the sake of anybody else that might read this other than myself, would anybody be kind enough to break the equation down into baby words please?

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  • Selection of a mesh with arbitrary region

    - by Tigran
    Considering example: I have a mesh(es) on the OpenGL screen and would like to select a part of it (say for delete purpose). There is a clear way to do the selction via Ray Tracing, or via Selection provided by OpenGL itself. But, for my users, considering that meshes can get wired surfaces, I need to implement a selection via a Arbitrary closed region, so all triangles that appears present inside that region has to be selected. To be more clear, here is screen shot: I want all triangles inside black polygon to be selected, identified, whatever in some way. How can I achieve that ?

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  • Correct way to use Farseer Physics in XNA

    - by user1640602
    I am using Farseer Physics for my 2D sidescroller game and I'm not sure how to proceed with it. I currently have a Sprite class (handles nothing but graphics), a GameObject class (contains specific object info like hit points), a World object which contains the list of Bodies, and a Level object which contains all of these objects. Originally I was trying to keep track of the Sprites, GameObjects, and Bodies separately because I felt that would provide loose coupling but it quickly became a headache. So my new idea was to add a Sprite member to the GameObject class but I'm still not sure how to maintain the Bodies because they have to communicate with GameObject. Specifically, my issue is this: The position of the Body is used to draw the Sprite inside of the Level. In order to do that I would have to maintain a link between GameObjects and Bodies. Is this correct or is there a better way to architect my game? If any of this is unclear please ask and I'll try to clarify. Thank you in advance for any help.

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  • How to Effectively Create Bullet Patterns

    - by SoulBeaver
    I'm currently creating a top-down shooter like Touhou. The most important factor of the game is that there are many diverse patterns and ways at which bullets are generated and shot at the player, see this video: http://www.youtube.com/watch?v=4Nb5Ohbt1Sg#start=0:60;end=9:53; At the moment, I'm using a class "Pattern" which has a series of steps on moving and shooting. However, I feel this method is quite laborous as I have to create a new Pattern for each attack and perhaps new Bullet classes that will implement a certain behavior. This question received a comment suggesting I should look into BulletML for easy creation and storage of bullets with a specific pattern. It looks decent, but it led me to wonder, what other solutions do you have that I should take into consideration? Update My current design is as follows: An example of an implemented pattern: My GigasPattern first executes a teleport which moves Alice to a certain point (X, Y) on the screen. After this is completed, the pattern starts using the Mover to move the sprite around (whereas teleporting has separate effects and animation). These are of no concern, really, as they are quite simple. The Shooter also creates various Attacks, which are classes again that the Shooter can use to create various patterns of bullets, much like the one in the question I posted. Once the Mover has reached it's destination, both it and the shooter stop and return to an inactive state. The pattern completes, is removed by the AI and a new one gets chosen.

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  • Calculating the rotational force of a 2D sprite

    - by Jon
    I am wondering if someone has an elegant way of calculating the following scenario. I have an object of (n) number of squares, random shapes, but we will pretend they are all rectangles. We are dealing with no gravity, so consider the object in space, from a top down perspective. I am applying a force to the object at a specific square (as illustrated below). How do I calculate the rotational angle, based on the force being applied, at the location being applied. If applied in the center square, it would go straight. How should it behave the further I move from the center? How do I calculate the rotational velocity?

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  • Where to start in creating a massive multiplayer 3D Java game [on hold]

    - by user1373771
    I am planning on creating a massive multiplayer world and I am wondering where to start. I am quite inexperienced in the field of Java but I have researched into it and learned that it is perhaps my best bet in creating this project is Java for the fact that it has a much easier learning curve than C++ to beginners and still capable of holding massive amounts of players at a time. My question is simple: Should I start the game by creating a single player prototype and introducing multiplayer later as I become more experienced or start with multiplayer before I am completely experienced in the field. Thanks for your help!

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  • How can i get almost pixel perfect collision detection in a multiplayer game?

    - by Freddy
    I'm currently working on a multiplayer game for iPhone. The problem i have, as with all multiplayer games, is that the other user will always see everything at a non-constant delay. The game I'm making need to have a almost pixel perfect collision detection, but 1 or 2 pixels off is not that big of a deal. How can I possibly get this working? I guess I could just set local player to also be at X ms delay. However this will probably just be worse and feel sloppy when the user input. I know this problem is probably something network programmers deal with everyday and I would be glad if someone could give me a possible solution for this.

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  • The most efficent ways for drawing lines all day long with OpenGL

    - by nkint
    I'd like to put a computer screen that is running an OpenGL programs in a room. It has to run all day long (not in the night). I'd like to draw lines that are slowly fading in the background. The setting is simple: a uniform color background (say, black) and colored lines (say, white) that are slowly fading out. With slowly I mean.. hours. Say that the first line I draw is with alpha 255 (fully visible), after one hours is 240. After 10 hours is 105. One line could have 250 points and there will be like 300 line in one day. For now I have done a prototype with very rudimentary method like: glBegin( GL_LINE_STRIP ); iterator = point_list.begin(); for (++iterator, end = point_list.end(); iterator != end; ++iterator) { const Vec3D &v = *iterator; glVertex2f(v.x(), v.y()); } glEnd(); More efficient method?

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  • Efficient Way to Draw Grids in XNA

    - by sm81095
    So I am working on a game right now, using Monogame as my framework, and it has come time to render my world. My world is made up of a grid (think Terraria but top-down instead of from the side), and it has multiple layers of grids in a single world. Knowing how inefficient it is to call SpriteBatch.Draw() a lot of times, I tried to implement a system where the tile would only be drawn if it wasn't hidden by the layers above it. The problem is, I'm getting worse performance by checking if it's hidden than when I just let everything draw even if it's not visible. So my question is: how to I efficiently check if a tile is hidden to cut down on the draw() calls? Here is my draw code for a single layer, drawing floors, and then the tiles (which act like walls): public void Draw(GameTime gameTime) { int drawAmt = 0; int width = Tile.TILE_DIM; int startX = (int)_parent.XOffset; int startY = (int)_parent.YOffset; //Gets the starting tiles and the dimensions to draw tiles, so only onscreen tiles are drawn, allowing for the drawing of large worlds int tileDrawWidth = ((CIGame.Instance.Graphics.PreferredBackBufferWidth / width) + 4); int tileDrawHeight = ((CIGame.Instance.Graphics.PreferredBackBufferHeight / width) + 4); int tileStartX = (int)MathHelper.Clamp((-startX / width) - 2, 0, this.Width); int tileStartY = (int)MathHelper.Clamp((-startY / width) - 2, 0, this.Height); #region Draw Floors and Tiles CIGame.Instance.GraphicsDevice.SetRenderTarget(_worldTarget); CIGame.Instance.GraphicsDevice.Clear(Color.Black); CIGame.Instance.SpriteBatch.Begin(); //Draw floors for (int x = tileStartX; x < (int)MathHelper.Clamp(tileStartX + tileDrawWidth, 0, this.Width); x++) { for (int y = tileStartY; y < (int)MathHelper.Clamp(tileStartY + tileDrawHeight, 0, this.Height); y++) { //Check if this tile is hidden by layer above it bool visible = true; for (int i = this.LayerNumber; i <= _parent.ActiveLayer; i++) { if (this.LayerNumber != (_parent.Layers - 1) && (_parent.GetTileAt(x, y, i + 1).Opacity >= 1.0f || _parent.GetFloorAt(x, y, i + 1).Opacity >= 1.0f)) { visible = false; break; } } //Only draw if visible under the tile above it if (visible && this.GetTileAt(x, y).Opacity < 1.0f) { Texture2D tex = WorldTextureManager.GetFloorTexture((Floor)_floors[x, y]); Rectangle source = WorldTextureManager.GetSourceForIndex(((Floor)_floors[x, y]).GetTextureIndexFromSurroundings(x, y, this), tex); Rectangle draw = new Rectangle(startX + x * width, startY + y * width, width, width); CIGame.Instance.SpriteBatch.Draw(tex, draw, source, Color.White * ((Floor)_floors[x, y]).Opacity); drawAmt++; } } } //Draw tiles for (int x = tileStartX; x < (int)MathHelper.Clamp(tileStartX + tileDrawWidth, 0, this.Width); x++) { for (int y = tileStartY; y < (int)MathHelper.Clamp(tileStartY + tileDrawHeight, 0, this.Height); y++) { //Check if this tile is hidden by layers above it bool visible = true; for (int i = this.LayerNumber; i <= _parent.ActiveLayer; i++) { if (this.LayerNumber != (_parent.Layers - 1) && (_parent.GetTileAt(x, y, i + 1).Opacity >= 1.0f || _parent.GetFloorAt(x, y, i + 1).Opacity >= 1.0f)) { visible = false; break; } } if (visible) { Texture2D tex = WorldTextureManager.GetTileTexture((Tile)_tiles[x, y]); Rectangle source = WorldTextureManager.GetSourceForIndex(((Tile)_tiles[x, y]).GetTextureIndexFromSurroundings(x, y, this), tex); Rectangle draw = new Rectangle(startX + x * width, startY + y * width, width, width); CIGame.Instance.SpriteBatch.Draw(tex, draw, source, Color.White * ((Tile)_tiles[x, y]).Opacity); drawAmt++; } } } CIGame.Instance.SpriteBatch.End(); Console.WriteLine(drawAmt); CIGame.Instance.GraphicsDevice.SetRenderTarget(null); //TODO: Change to new rendertarget instead of null #endregion } So I was wondering if this is an efficient way, but I'm going about it wrongly, or if there is a different, more efficient way to check if the tiles are hidden. EDIT: For example of how much it affects performance: using a world with three layers, allowing everything to draw no matter what gives me 60FPS, but checking if its visible with all of the layers above it gives me only 20FPS, while checking only the layer immediately above it gives me a fluctuating FPS between 30 and 40FPS.

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  • Confusion about Rotation matrices from Euler Angles

    - by xEnOn
    I am trying to learn more about Euler Angles so as to help myself in understanding how I can control my camera better in the game. I came across the following formula that converts Euler Angles to rotation matrices: In the equation, I could see that the first matrix from the left is the rotation matrix about x-axis, the second is about y-axis and the third is about z-axis. From my understanding about ordinary matrix transformations, the later transformation is always applied to the right hand side. And if I'm right about this, then the above equation should have a rotation order starting from rotating about z-axis, y-axis, then finally x-axis. But, from the symbols it seems that the rotation order start rotating about x-axis, then y-axis, then finally z-axis. What should the actual order of the rotation be? Also, I am confuse about if the input vector, in this case, would be a row vector on the left, or a column vector on the right?

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  • The right way to add images to Monogame/Windows

    - by ashes999
    I'm starting out with MonoGame. For now, I'm only targeting Windows (desktop -- not Windows 8 specifically). I've used a couple of XNA products in the past (raw XNA, FlatRedBall, SilverSprite), so I may have a misunderstanding about how I should add images to my content. How do I add images to my project? Currently, I created a new Monogame project, added a folder called "Content," and added images under there; the only caveat is that I need to set the Copy to Output Directory action to one of the Copy ones. It seems strange, because my "raw" XNA project just last week had a Content project in it (XNA Framework Content Pipeline, according to VS2010), which compiled my images to XNB (I think). It seems like Monogame doesn't use the same content pipeline, but I'm not sure. Edit: My question is not about "how do I get the XNA content pipeline to work with Monogame." My question is "why would I want to use the XNA content pipeline in Monogame?" Because there are (at least) two solutions (that I see today): Add the images to the Monogame project and set the Copy to Output Directory options to copy. Add a XNA content pipeline project and add my images to that instead; reference it from my MOnogame project. Which solution should I use, and why? I currently have a working version with the first option.

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  • Working out of a vertex array for destrucible objects

    - by bobobobo
    I have diamond-shaped polygonal bullets. There are lots of them on the screen. I did not want to create a vertex array for each, so I packed them into a single vertex array and they're all drawn at once. | bullet1.xyz | bullet1.rgb | bullet2.xyz | bullet2.rgb This is great for performance.. there is struct Bullet { vector<Vector3f*> verts ; // pointers into the vertex buffer } ; This works fine, the bullets can move and do collision detection, all while having their data in one place. Except when a bullet "dies" Then you have to clear a slot, and pack all the bullets towards the beginning of the array. Is this a good approach to handling lots of low poly objects? How else would you do it?

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  • Simple 2D Flight Physics with Box2D

    - by MarkPowell
    I'm trying to build a simple side scroller with an airplane being the player. As such, I want to build simple flight controls with simple but realistic-feeling physics. I'm making use of cocos2D and Box2D. I have a basic system working, but just can't get the physics feeling correct. I am applying force to the plane (which is a b2CircleShape) based on the user's input. So, basically, if the user pushes up, body_->ApplyForce(b2Vec2(10,30), body_->GetPosition()) is called. Similarly, for down -30 is used. This works and the plane flys along with up/down causing it to dive or climb. But it just doesn't feel right. There is no slowdown on climbs, nor speed up during dives. My simple solution is far to simple. How can I get a better feel for a plane climbing/diving? Thanks!

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  • A way to store potentially infinite 2D map data?

    - by Blam
    I have a 2D platformer that currently can handle chunks with 100 by 100 tiles, with the chunk coordinates are stored as longs, so this is the only limit of maps (maxlong*maxlong). All entity positions etc etc are chunk relevant and so there is no limit there. The problem I'm having is how to store and access these chunks without having thousands of files. Any ideas for a preferably quick & low HD cost archive format that doesn't need to open everything at once?

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  • Very basic OpenGL ES 2 error

    - by user16547
    This is an incredibly simple shader, yet I'm having a lot of trouble understanding what's wrong with it. I'm trying to send a float to my fragment shader. Its purpose is to adjust the alpha of the fragment colour. Here is my fragment shader: precision mediump float; uniform sampler2D u_Texture; uniform float u_Alpha; varying vec2 v_TexCoordinate; void main() { gl_FragColor = texture2D(u_Texture, v_TexCoordinate); gl_FragColor.a *= u_Alpha; } and below is my rendering method. I get a 1282 (invalid operation) on the GLES20.glUniform1f(u_Alpha, alpha); line. alpha is 1 (but I tried other values as well) and transparent is true: public void render() { GLES20.glUseProgram(mProgram); if(transparent) { GLES20.glEnable(GLES20.GL_BLEND); GLES20.glBlendFunc(GLES20.GL_SRC_ALPHA, GLES20.GL_ONE_MINUS_SRC_ALPHA); GLES20.glUniform1f(u_Alpha, alpha); } Matrix.setIdentityM(mModelMatrix, 0); Matrix.rotateM(mModelMatrix, 0, angle, 0, 0, 1); Matrix.translateM(mModelMatrix, 0, x, y, z); Matrix.multiplyMM(mMVPMatrix, 0, mViewMatrix, 0, mModelMatrix, 0); Matrix.multiplyMM(mMVPMatrix, 0, mProjectionMatrix, 0, mMVPMatrix, 0); GLES20.glUniformMatrix4fv(u_MVPMatrix, 1, false, mMVPMatrix, 0); GLES20.glBindBuffer(GLES20.GL_ARRAY_BUFFER, vbo[0]); GLES20.glVertexAttribPointer(a_Position, 3, GLES20.GL_FLOAT, false, 12, 0); GLES20.glBindBuffer(GLES20.GL_ARRAY_BUFFER, vbo[1]); GLES20.glVertexAttribPointer(a_TexCoordinate, 2, GLES20.GL_FLOAT, false, 8, 0); //snowTexture start GLES20.glActiveTexture(GLES20.GL_TEXTURE0); GLES20.glBindTexture(GLES20.GL_TEXTURE_2D, textureHandle[0]); GLES20.glUniform1i(u_Texture, 0); GLES20.glBindBuffer(GLES20.GL_ELEMENT_ARRAY_BUFFER, ibo[0]); GLES20.glDrawElements(GLES20.GL_TRIANGLE_STRIP, indices.capacity(), GLES20.GL_UNSIGNED_BYTE, 0); GLES20.glBindBuffer(GLES20.GL_ARRAY_BUFFER, 0); GLES20.glBindBuffer(GLES20.GL_ELEMENT_ARRAY_BUFFER, 0); if(transparent) { GLES20.glDisable(GLES20.GL_BLEND); } GLES20.glUseProgram(0); }

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  • Unity: Assigning String value in inspector

    - by Marc Pilgaard
    I got an issue with Unity I can't seem to comprehend, and it is possibly very simple: I am trying to write a simple piece of code in JavaScript where a button toggles the activation of a shield, by dragging a prefab with Resources.load("ActivateShieldPreFab") and destroying it again (Haven't implemented that yet). I wish to assign this button through the inspector, so I have created a string variable which appears as intended in the inspector. Though it doesn't seem to register the inspector input, even though I changed the value through the inspector. It only provides the error: "Input Key named: is unknown" When the button name is assigned within the code, there is no issues. Code as follows: var ShieldOn = false; var stringbutton : String; function Start(){ } function Update () { if(Input.GetKey(stringbutton) && ShieldOn != true) { Instantiate(Resources.load("ActivateShieldPreFab"), Vector3 (0, 0, 0), Quaternion.identity); ShieldOn = true; } } Hope somebody can help, in advance... Thanks

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  • What's the best way to move cars along roads

    - by David Thielen
    I am implementing car movement game (sort-of like Locomotion). So 60 times a second I have to advance the movement of each car. The problem is I have to look ahead to see if there is a slower car, stop sign, or red light ahead. And then slow down appropiately. I also want to have the cars take time to go from stopped to full speed and again to slow down. I'm not implementing full-blown physics, but just a tick by tick speed up/slow down as that provides most of the realism to match what people expect to see. The best I've come up with is to walk out the full distance the car would travel of it was slowing to a stop and see if anywhere along that path it needed to slow down or stop. And then move it forward appropiately. I am moving the cars 60 times a second so I need this to be fast. And walking out that whole path each tick strikes me as processor intensive. What's the best way to do this?

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  • Calculating Hit Accuracy score in a game

    - by N0xus
    I'm currently in the process of making a scoreboard for my game. One of things I would like to display is the players accuracy in the amount of hits they had in game. However, I have never done this before and I've no idea how to go about doing this. Is there a commonly used algorithm out there that can help me calculate this, or has someone found a way to calculate this fairly easily? Any help with this would be appreciated.

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  • Understanding dot notation

    - by Starkers
    Here's my interpretation of dot notation: a = [2,6] b = [1,4] c = [0,8] a . b . c = (2*6)+(1*4)+(0*8) = 12 + 4 + 0 = 16 What is the significance of 16? Apparently it's a scalar. Am I right in thinking that a scalar is the number we times a unit vector by to get a vector that has a scaled up magnitude but the same direction as the unit vector? So again, what is the relevance of 16? When is it used? It's not the magnitude of all the vectors added up. The magnitude of all of them is calculated as follows: sqrt( ax * ax + ay * ay ) + sqrt( bx * bx + by * by ) + sqrt( cx * cx + cy * cy) sqrt( 2 * 2 + 6 * 6 ) + sqrt( 1 * 1 + 4 * 4 ) + sqrt( 0 * 0 + 8 * 8) sqrt( 4 + 36 ) + sqrt( 1 + 16 ) + sqrt( 0 + 64) sqrt( 40 ) + sqrt( 17 ) + sqrt( 64) 6.3 + 4.1 + 8 10.4 + 8 18.4 So I don't really get this diagram: Attempting with sensible numbers: a = [1,0] b = [4,3] a . b = (1*0) + (4*3) = 0 + 12 = 12 So what exactly is a . b describing here? The magnitude of that vector? Because that isn't right: the 'a.b' vector = [4,0] sqrt( x*x + y*y ) sqrt( 4*4 + 0*0 ) sqrt( 16 + 0 ) 4 So what is 12 describing?

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