Working With Triangle Meshes

1. Introduction

So what, exactly, are triangle meshes? The most obvious answer is, "a surface composed of triangular facets." In most 3D programs, that would also be the complete answer. In Art of Illusion, it is only the beginning.

A more accurate answer is, "a surface defined by a mesh of triangular facets." Note the difference: "defined by" rather than "composed of". The triangular facets act as a control mesh by which you can define the surface. The surface itself, however, may be either faceted or smoothly rounded. Or half one and half the other. Or mostly smooth, but with a few sharp points and creases. Or anything in between.

This tutorial will introduce you to triangle meshes: what they are, how to create them, and some of the tools available for editing them. It is not intended to be comprehensive: I will not be covering every last feature of the mesh editor. I will, however, try to get most of them, and the remaining ones should be easy enough to figure out.

Before going on to the tutorial itself, I should mention two other types of objects that can also be used for creating free-form surfaces: spline meshes and polymeshes. Spline meshes lend themselves more naturally to creating certain types of shapes than do triangle meshes, but they are much less powerful overall. For this reason, you will sometimes want to use spline meshes for creating certain types of fairly simple objects, but not for anything which is very complex. Polymeshes, on the other hand, are in some ways more powerful than triangle meshes, in that they allow faces to have more than three sides. This makes them very useful for many types of objects, although they are less efficient to render and in some cases produce a less smooth surface.

First, every triangle mesh has a smoothing method, which determines how the actual surface is calculated from the control mesh. There are four different methods to choose from, which are illustrated in the following figure.

A.	B.
C.	D.

• None If you choose this method (shown in A), the surface will be identical to the control mesh. That is, it will be a simple mesh of flat, triangular facets.
• Shading This method causes the triangles to be shaded in such a way that they seem to blend smoothly together. If you look at B, you will see that the surface no longer looks faceted. It looks like a smooth surface with no sharp edges. On the other hand, if you look at the object's silhouette, you will see that it really is still made of sharp edges and corners. It simply has been shaded in a way that hides this fact.
• Interpolating This method actually changes the shape of the surface to eliminate the edges and corners. It calculates a smooth (C1 for the mathematically inclined) surface which passes through the vertices of the control mesh. If you look at the object shown in C, you will see that it has a rounded profile with no corners.
• Approximating Like the interpolating smoothing method, this method creates a smoothly rounded surface based on the control mesh. In this case, however, the surface is not required to actually pass through the vertices of the control mesh, which allows it to be even smoother. Compare the objects in C and D, and you will see the difference.

Triangle meshes also offer a second tool which you can use to control the shape of the surface: every vertex and every edge has a smoothness value associated with it. A value of 0 produces a sharp point or crease. A value of 1 gives a smooth surface. Values between these extremes give intermediate results: rounded points, beveled edges, etc.

The following figure gives you an idea of what can be done just by changing the smoothness values of a mesh. All of the images show the same object: a cube which was converted to a triangle mesh, and whose smoothing method was set to "approximating". The only difference between the images is the smoothness values for vertices and edges.

For some objects (like cubes), it is possible to create a triangle mesh which exactly reproduces the shape of the original object. In most cases, however, the mesh will only be able to approximate the original surface. In these cases, after you select the "Convert to Triangle Mesh" command, you will be asked to specify a tolerance for the mesh. This is the maximum distance by which the mesh can deviate from the original object. (One distance unit corresponds to 100 pixels on the screen.) That is, no point on the (unsmoothed) mesh will ever be further than the specified tolerance from the original surface. Using a small tolerance will guarantee that the triangle mesh closely reproduces the original object, but may result in a very complex mesh containing a very large number of triangles.

In most cases, it is better to use a larger tolerance. Simple meshes are easier to edit and faster to render. Furthermore, by taking advantage of the mesh smoothing features described in the previous section, even a low accuracy mesh can closely reproduce the appearance of the original object.

This is illustrated in the following figure. On the left is a cylinder object. Being an exact mathematical cylinder, its sides are smooth and perfectly round.

This cylinder was then converted to a triangle mesh with a large tolerance. This is shown in the center, with the smoothing method set to "none". It is clear from the image that the mesh is only a very rough approximation to the original cylinder.

The triangle mesh editor window is quite similar to the main scene editor window. The four viewports in the center show the object that is being edited (both the control mesh, in black, and the actual surface, in blue). The tool palette along the left edge provides various tools for editing the mesh, and the line of text at the bottom describes how to use the current tool.

Two of the tools should already be familar to you from the scene editor: Move View and Rotate View. These tools merely change the direction from which you are looking at the object, and do not affect the object itself in any way.

The most important tool is the Select and Move tool. As the name suggests, this tool is used for two purposes: selecting portions of the mesh to edit, and moving around selected vertices. You can select vertices by clicking on them, select several vertices by shift-clicking, or drag a box to select everything inside it. Clicking on a vertex and then dragging will move all the selected vertices.

The next five tools (Scale, Rotate, Skew, Taper, and Outset) are all used to deform pieces of the mesh in specific ways. To use any of these tools, you must first select a portion of the mesh to deform. It will then display a set of handles around the selection, which you can drag to deform the selected region. If you experiment with the various tools, you will quickly get a feel for what each one does. Be sure to read the text at the bottom of the screen, since it will tell you about various keys you can hold down that affect the behavior of the tool.

The multicolored sphere icon represents the compound Move/Scale/Rotate tool. As the name suggests, it combines most of the functions of the Move, Scale, and Rotate tools into a single tool. Its user interface is a little more complicated than the others, but once you get comfortable with it, it can be very convenient to use, since it lets you do many operations without having to change tools.

The two light blue icons are the Bevel/Extrude and Create Point tools. They are discussed in sections 6 and 7 below.

The green stick figure icon represents the Skeleton tool. It is used for creating and editing the mesh's skeleton, and will be discussed in detail in section 9.

In the previous paragraphs, I referred to selecting and moving vertices. Actually, the editing tools are a little more general than that. In the lower left corner of the window, you will notice three buttons labeled "Point", "Edge", and "Face". These are the three selection modes supported by the mesh editor. In "Edge" mode, the basic objects which you can select and manipulate are the edges of the mesh (the lines connecting vertices). In "Face" mode, the basic objects are faces. Try experimenting with all three modes. Also notice what happens to your current selection when you switch modes.

This can be a very difficult thing to do. Trying to reshape the surface by moving just one vertex at a time is both difficult and tedious. You can select a group of vertices to move all at once, but this tends to create a sharp ridge at the edge of the selected region.

The "Mesh Tension" command can be a huge help at these times. It causes the mesh to behave as if it were elastic: as you drag one vertex, nearby vertices will be pulled after it so that the surface remains smooth.

To use this feature, select "Mesh Tension" from the Edit menu. A window will appear with two options: Maximum Distance and Tension. Maximum Distance sets the radius of effect of the tension. It is measured in edges. If you set it to 0 (the default), only selected vertices will move as you drag them. If you set it to 1, unselected vertices which share an edge with a selected vertex will also move (though not as much as the selected ones). If you set it to 2, vertices which are two edges distant from the selection will move, and so on. In all cases, the further a vertex is removed from the selection, the less it will move.

Suppose you are trying to create a head for a character. You might start with a sphere (a reasonable first approximation to the shape of a head), then convert it to a triangle mesh to refine the shape: square the forehead, pull out the jaw, etc. Suppose that when you come to do the face, however, you find that your mesh is too low resolution. There simply are not enough vertices in it to let you sculpt a realistic nose, mouth, eyes, and so on. What can you do?

The answer is to use the "Subdivide" command. This command adds detail to your mesh by subdividing the selected region to increase the number of vertices, faces, and edges. The precise way in which it does this depends on your current selection mode:

• In "Edge" mode, it adds a new vertex in the middle of each selected edge. The edge itself is split into two edges, and the adjoining faces are similarly split.
• In "Face" mode, it adds a new vertex in the center of each selected face. New edges are added connecting this vertex to each of the corners of the face, and the face itself is split into three faces.

If this sounds very confusing, simply remember this rule: if you are in Edge mode and you repeatedly use the Subdivide command, the control mesh will quickly converge toward the actual smoothed surface of the object. Try it a few times and you will see what I mean.

In addition to the "Subdivide" menu item, there is also the Create Point tool which does the same thing in a more interactive way. With this tool selected, you can click on an edge or face (depending on which selection mode you are in), and it will be split at exactly the point where you clicked.

By using the Subdivide command, you can create large numbers of new vertices and edges very quickly. This itself can be a problem. Remember what I said before about simple meshes being easier to work with. Once you have finished subdividing and editing a part of the mesh to create the shape you want, you often will find that it has more vertices than are actually required. After a few cycles of editing, this can become a serious problem.

The solution is to use the "Simplify" command. This does the opposite of the Subdivide command: it merges adjacent vertices together to reduce the total number of vertices, edges, and faces in the mesh. To use it, select the portion of the mesh you wish to simplify and select "Simplify Selection" from the Mesh menu. Alternatively, if nothing is selected, the command changes to "Simplify Mesh", which simplifies the entire mesh.

The simplification is done based on a local error metric. This means that you are prompted to enter a maximum allowed error, and the control mesh is then simplified as far as possible while ensuring that no point on the new, simplified mesh is further than that distance from the original, unsimplified mesh. Small values ensure that the simplified mesh remains very close in shape to the original mesh, but also limit how much it can be simplified. Larger values result in a simpler final mesh, but may cause noticeable changes to the shape of the surface.

A window will appear in which you can specify details of how the operation should be performed. You can set the following options:

• Extrude Height This specifies how far the extrusions extend outward from the surface.
• Bevel Width This adjusts the shape of the extrusions. A positive value causes them to taper inward, so that their tips are smaller than their bases. A negative value causes them to taper outward, so the tips are larger than the bases.
• Apply To This option is only available in Face mode. It can be set to either "Individual Faces" or "Selection as a Whole". When applied to individual faces, there will be a separate extrusion from each selected face. When applied to the selection as a whole, then all selected faces will be extruded as a group (or, more accurately, there will be one extrusion for each contiguous group of selected faces). The following figures illustrate the difference between the two choices.

There also is a Bevel/Extrude tool which provides a more interactive (though less precise) interface for doing the same thing. To use it, select a portion of the mesh and then drag the handle shown in the center of the selected region. You can move the mouse up and down to change the extrude height, and left and right to change the bevel width.

The first such command is the "Clear" command in the Edit menu. This deletes the selected vertices, edges, or faces from the mesh, leaving a hole where they were. You can also do this by pressing the Delete key on your keyboard.

The "Close Selected Boundary" command does the exact opposite: it creates a new set of faces to close off a hole in the mesh. To use it, make sure you are in Edge selection mode, then select the edges which surround the hole you want to close off. (You must select all edges surrounding the hole; if there is a break anywhere in the selection, the menu item will be disabled.)

The "Join Selected Boundaries" command works in a similar way, but instead of closing off a single hole, it creates faces that connect two holes in different parts of the mesh. To use it, select the boundary curves of both holes that you want to connect and select the command from the Mesh menu.

To connect the two holes, an edge must be added linking each vertex of one boundary to a vertex of the other boundary. There are, of course, many ways to do that. The editor tries to make a good guess about how best to connect the holes, but it will not always do it exactly the way you want. It therefore displays a dialog with a preview of how the object will look after joining the boundaries. You can use the controls in this dialog to edit how the two boundaries should be connected together. When you are satisfied with the way it looks, click OK to complete the operation.

Skeletons are a useful tool for creating meshes, but when you come to animation they become absolutely critical. There are two specific ways in which skeletons help you to animate objects:

1. They allow you to create keyframes quickly and easily. For example, if you want a character to raise its left arm, you simply select the bone corresponding to the arm and raise it. There is no need to select all the vertices which make up the arm and try to rotate them in a realistic way.
2. When Art of Illusion interpolates between the keyframes you have set, it will use information about the object's skeleton to interpolate in a more physically realistic way than it could otherwise.

The figure above shows a simple mesh, and its skeleton consisting of three bones. The bones are the diamond shaped objects, and the joints are the crosses at the ends of them.

The Skeleton tool is very easy to use. You select a joint by clicking on it, and drag to move it. The currently selected joint is colored red, as shown in the figure above. To create a new bone, click the mouse with the control key held down. This will create a new joint at the place you clicked, and a new bone connecting it to the previously selected joint. If no joint is currently selected, the new joint will not be connected to any bone. This allows you to create skeletons consisting of several separate groups of bones, with no connection between one group and another.

Notice that one of the joints in the skeleton is colored green. This joint has been locked. You can lock or unlock any joint by shift-clicking on it. When you drag any other joint, a locked joint (and everything beyond it) will remain fixed, and the bones in between will bend to follow your movements. Try creating a skeleton, and experiment with moving the pieces of it around. You will quickly get a feel for how it works. If you want to delete a joint, simply select it and press the delete key.

Notice the red handle coming out from the side of the selected joint. If you drag it, the selected joint will bend without changing any other joint. This can be useful when you want to make precise adjustments in the shape of the skeleton. A handle is drawn for every unlocked degree of freedom of the selected joint. I'll explain in the next section exactly what that means.

Now you know how to create and move a skeleton, but that isn't very useful by itself. It's the mesh that you really want to reshape, and moving the skeleton doesn't seem to have any effect on it so far. To make that happen, you must bind the mesh to the skeleton. Here is how you do that:

• Choose "Select All" from the Edit menu.
• Choose "Bind Points to Skeleton..." from the Skeleton menu.
• Click "OK", accepting the default value for IK Weight Blending.

Select a vertex in your mesh and choose "Edit Points..." from the Mesh menu. You will notice two items in the window called IK Bone and IK Weight. IK stands for inverse kinematics, which is the name for the way the skeleton bends to follow your movements as you drag a joint around the screen.

The first option allows you to specify which bone of the skeleton this particular vertex is bound to. Any time that bone moves, the vertex will move with it. By repeatedly selecting vertices, choosing "Edit Points...", and setting bones for them, you can manually bind the mesh to the skeleton. You usually will not need to, since the "Bind Points to Skeleton..." command does this for you automatically.

If you simply bound each vertex to a bone, the results would not be very satisfying. When you adjusted the skeleton, sharp creases or distortions would appear in the mesh around every joint, as the vertices bound to one bone moved in a different direction from those bound to the next bone over. To reduce this problem, a vertex can be "partially bound" to two bones at once. That is what the IK weights are for. If you bind a vertex to a bone with a weight of 1, it is completely bound to that bone. As you reshape the skeleton, its position will be entirely determined by the motion of that bone. On the other hand, if you set its weight to 0, it will not be bound to that bone at all. Instead, it will be entirely bound to that bone's parent in the skeleton. And if you assign a weight between 0 and 1, it will be partly bound to each of the two bones, and its motion will be determined by a weighted average of their respective positions. By using intermediate weights around each joint, you can smooth out the distortions in the mesh and create much more satisfactory results.

Most of the time, you do not need to worry about any of this. The "Bind Points to Skeleton..." command generally does a fairly good job of figuring out which bone to attach each vertex to and what weight to use. There may be times, however, when you are not satisfied with its assignments, and in these cases you will want to adjust them by hand.

One other command that is very useful is "Temporarily Detach Skeleton" in the Skeleton menu. Suppose you have carefully bound your mesh to its skeleton, adjusting weights until you are satisfied with the result. Then you decide you want to change the position of the skeleton slightly without affecting the mesh. You could select all the points, set them to be not bound to any bone, move the skeleton, and then manually reattach them again. That would be a huge amount of work, however. Instead, you can simply select "Temporarily Detach Skeleton". Whenever this option is selected, moving the skeleton has no effect on the mesh regardless of whether points are bound to it or not. When you are done moving the skeleton, you can simply deselect the option again.

Now let's look at a single bone in more detail. Select a bone and choose the "Edit Bone..." command from the Skeleton menu. (Of course, it is really a joint you are selecting, but this is one of those cases where things make far more sense if you think in terms of selecting bones, not joints. For example, if you want to control how a character's arm bends at the elbow, you do not select the elbow, but instead select the joint at the end of the arm. This seems bizarre if you think in terms of joints, but makes sense if you think in terms of bones. You want to control the motion of the forearm. Therefore you want to select the forearm, which means clicking on the joint at its end.) A window will appear which looks something like this:

Who would have guessed that a single bone could have so much information to display and edit! The window is divided into four sections, one for each of the bone's four degrees of freedom. A degree of freedom is a way in which the bone can move or change.

The first two, X Bend and Y Bend, describe the orientation of the bone relative to its parent bone. To understand these, hold your arm pointing straight out to one side. Move the arm slowly up and down. You are changing its X Bend angle. Now move it slowly forward and backward. You are changing its Y Bend angle. Make sense?

The third degree of freedom, Twist, describes a rotation of one end of the bone relative to the other around the axis of the bone. To understand this, hold your arm straight out in front of you with the palm facing down. Now rotate your hand so the palm faces upward. That is your forearm's twist degree of freedom. Notice how one end of your forearm (the end connected to your wrist) rotates by 180 degrees, while the other end (the end connected to your elbow) hardly moves at all.

Of course, real bones are solid objects whose ends cannot rotate relative to each other. Your forearm actually contains two different bones which rotate around each other to create the twist of the arm as a whole. For our purposes, however, it is much easier to use only a single bone and allow it to have an internal twist.

The final degree of freedom is Length. Once again, real bones are solid objects which cannot change in length. But why should you be bound by reality?

So what are all of those options for each degree of freedom? First and most importantly is the actual value for the angle (or distance in the case of length). You can edit it by typing in a new value, or by dragging the handle on the dial next to it.

Next there is the "Lock" checkbox, which allows you to lock a particular degree of freedom so that it cannot be changed. Each degree of freedom represents a way in which the bone can move, and different bones move in different ways. For example, your upper arm can rotate up and down (X Bend) and forward and back (Y Bend), but it cannot twist or change length. Your forearm can bend at the elbow in one direction (X Bend) but not in the other direction (Y Bend). Imagine how strange it would look if an animated character had elbow joints that could bend in any direction, just like a shoulder joint!

Next, you can restrict the range of motion for a degree of freedom. For example, your forearm can go from being completely straight (0 degrees) to being quite sharply bent inward (perhaps 150 degrees). But it cannot bend inward any further than that, nor can it bend outward at all. Setting ranges for each degree of freedom is important for making an animated character move in a realistic fashion.

If you want still more control over the motion of a degree of freedom, you can set a comfort range for it. This is a subset of its allowed range over which it moves most easily. As it moves outside its comfort range, it becomes stiffer and moves less easily. Thus, although it can still move all the way to the end of its allowed range, it will usually tend to remain within the comfort range.

Finally, you can set an overall stiffness for each degree of freedom which determines how easily it moves. Suppose, for example, you are modelling an animal with a long tail, and you place a series of several bones along the length of the tail so that it can bend freely. To bend the tail, you want to simply lock the joint at the base of the tail, select the end of the tail, and drag it. You may find that you are not satisfied with the way it bends, however. Perhaps it bends too much at the end and not enough near the base. You can fix this by increasing the stiffness of the joints near the end. Or perhaps it bends too much near the base and not enough at the end. In that case, you would increase the stiffness of the joints near the base.

If all of these options sound a little overwhelming, don't worry! You can safely ignore most of them, especially the stiffness values and comfort ranges. They are a handy tool once you get more comfortable working with skeletons, but they are only that: a tool which you can use or not as you choose.