NURBS Surfaces

Many of the direct curve manipulation tools, such as Break and Sharpen, can also be applied to surfaces. The only difference is that surfaces have two parameter directions  (called 'u' and 'v') in which they can be manipulated.

In this tutorial, we will demonstrate these direct NURBS mesh manipulation tools.
 

Creating basic NURBS objects

The three basic creation tools can be used to create NURBS surfaces with freeform, circular and rectangular shapes.

To create NURBS objects using the Unified tools, set the 'Class' option to 'Geometry' and the 'Type' option to 'NURBS'. However, to get a surface rather than a curve, set the 'Filled' and/or 'Extruded' options.

Let's create a rectangular cubic NURBS mesh, which consists of 4 x 4 control points:

1. Activate the Unified/Rectangle tool.

As always, the control bar shows you the tool specific options.

Make sure the 'Type' option is set to 'NURBS'. Set 'Order' to cubic, check 'Filled' and specify the control point count 4x4 through the 'U' and 'V' fields.

The unified Rectangle tool activated. Options for creating a 4x4 cubic mesh.

2. Then enter two points through a view window to define geometry for the rectangle.

A rectangular NURBS mesh consisting of 4x4 control points
 

Curve handles

To switch to the curve editing mode:

1. Select the mesh. As usual, the control bar shows you the tools, which can be applied to the selected mesh.

NURBS mesh specific tools

2. In the control bar, set the 'Edit' control to 'Curves'. Just click on the 'Edit' icon and select 'Curve' from the drop down menu. The mesh now shows you two vertical and two horizontal curve handles. When you move the mouse over a curve handle, the color of the curve is changed. As usual, you can move any curve simply by dragging it with the mouse.

The left vertical curve selected and moved upwards

You can also multi select any number of curves and apply any transformation or deformation tool to them.
 

Control point handles

To enter the control point editing mode, set the 'Edit' control in the control bar to 'Point' to access the control point handles.

Hint: use 'compass' menus to switch between different editing modes easily.

Just drag the control points, using the modification handles with the mouse to move them.

Control point handles shown.
 

Adding new curves

Just like you can add a new point to a curve, you can add a new curve to a mesh.

1. Use the Unified/Rectangle tool to create a planar cubic NURBS mesh consisting of 4x4 control points.

2. Select the 'Add V Directed Curve' pop-up menu item of the 'Add' tool.

Activate the 'Add Curve' tool with the 'V directed curve' option

Hint: you can also use the compass menu to activate the tool.

2. Now press down the left mouse button over the mesh. Move the mouse while still holding down the left mouse button. The tool renders the new curve over the mesh. When a desired position is found, just release the mouse.

Activate the 'Add' tool and LMB drag over the mesh to find a desired position for the new curve

Note: the shape of the mesh is not affected by the new curve. So it does not make any sense to add new curves without modifying them.

The new curve moved and scaled

Similarly, you can add a new horizontal curve to the mesh. Just select 'Add U directed curve' from the pop-up menu of the 'Add' tool and drag over the mesh to define the position for the curve.

A new 'U' directed curve added and moved

Most of the direct mesh manipulation tools work this way.
 

Creating sharp curves

The 'Sharpen' tool works similar to the 'Add' tool. The only difference is that the inserted curve breaks the continuity of the mesh so that a sharp edge can be formed.

Let's imagine we need to add a 'sharp' V directed curve to the mesh we created above.

1. Select the 'Add V directed curve' of the 'Sharpen' tool and drag over the mesh to specify a position for the curve. The workflow is exactly the same as demonstrated in the previous example.

The 'Sharpen' tool doesn't change the shape of the mesh. However, the nature of the new inserted curve is revealed as soon as you modify the mesh. For example, move the rightmost vertical curve upwards a bit.

Sharp 'v directed' curve added and the rightmost curve moved upwards
 

Breaking meshes

The 'Break' tool breaks the mesh through the specified curve. Again, the workflow is similar to the previous 'Add' and 'Sharpen' tools.

Let's split the mesh we created in the previous example into four parts:

1. Click the 'Break' tool button.

2. Drag over the mesh to define the positions for the 'u' and 'v' directed curves.

The break tool activated with both 'u' and 'v' options

As you can see, two perpendicular curves passing through the specified point are rendered over the mesh. The reason for this is that clicking the 'Break' button selects both 'u' and 'v' options. This is also true for 'Add' and 'Sharpen' tools.

When a desired position for the curves is found, just release the mouse and the mesh is split into four parts by the curves.

Pieces moved apart

If the mesh is closed in the given direction, the specified curve just opens the mesh through the specified curve. Let's demonstrate this by applying the break tool to a NURBS sphere. The surface of a sphere is closed in its 'u' direction but open in its 'v' direction.

1. Switch to the top view and create a cubic NURBS sphere by using the NURBS/Sphere tool.

2. Switch to the front view and select 'Break by U directed curve' pop-up menu item of the 'Break' tool.

Activate the 'Break' tool with the 'Break by U directed curve' option

3. Drag over the sphere to find a desired break curve. When found, release the mouse to accept the tool.

North pole cut off

Because the 'u' directed curve forms a closed loop (the mesh is closed in its 'u' direction), the tool split the surface into two separate parts. Correspondingly, the result is two separate NURBS meshes.

So, what happens if we break the mesh in its 'v' direction? In that direction the surface is open: All 'v' directed curves have end points at the poles of the sphere.

4. Select the north-pole mesh and activate 'Break by V directed curve'. Then drag the mouse over the semi sphere to find a desired position for the break curve. When found, release the mouse.

As you can see, new objects were not created. Also, the shape of the mesh remains exactly the same. However, the mesh was opened through the specified curve, as you can see by modifying the mesh.

Semi sphere broken by a 'v' directed curve and edited

Now the semi sphere is open in the both parameter directions. Let's apply 'Break by V directed curve' again.

Now the result is two separate meshes.
 

Defining 'Start point' of a closed mesh

Just like you can redefine the start point for a closed curve, you can redefine the start curve for a closed surface. Again, the only difference to the curve start tool is that now there are two parameter directions 'u' and 'v' in which the start curve can be defined.
 

Weights

NURBS meshes support a parameter called 'weight'.

The weight parameter controls how strongly the control point in question affects the corresponding surface point (knot point).  The higher the weight, the stronger the control point pulls the surface towards the control point.

You can modify the weight parameter interactively as follows:

1. In 'Edit' mode, select the desired control points.

2. Select the 'Weight' tool and click the mouse in a view window. By moving the mouse up/down you can increase/decrease the weight of the selected control points. Click the mouse again to accept the tool.

A high weight value set to three selected control points

Note: You can also modify the weight of the selected control points by dragging them while holding down the Alt key.
 

Degree elevation/reduction

The Degree Elevation and Degree Reduction tools modify the order (degree) of the mesh. We already demonstrated the effect of degree elevation/reduction in the context of NURBS curves. The only difference to NURBS curves is that NURBS meshes have two directions in which the order can be controlled independently.

A mesh, which is cubic in 'u' direction and polygonal in 'v' direction
 
Polygonal, quadric and cubic meshes are sufficient for most practical applications. Surfaces with higher degree are sometimes needed when an extremely high surface smoothness and rendering quality is desired.
 

Concatenating meshes

The 'Concatenate' tool concatenates two meshes with each other provided that the meshes have an edge i.e. they are open at least in one parameter direction. The tool concatenates the two nearest edges.

To use the concatenate tool, move the meshes to be concatenated so that the edges, which should be connected with each other, are placed side by side. The tool finds the nearest edges and connects them.

 
Original meshes on the left. Concatenate tool applied on the right.
 

Swap

Sometimes you have to swap the direction of a mesh. For example, it may be necessary to invert the direction of a mesh to get it concatenated properly to another mesh.

The 'Swap' tool swaps the direction in both 'u' and 'v' directions. To swap the direction in one direction only, select the appropriate pop-up menu item from the tool button.

Swap the selected mesh in its 'u' direction
 

Controlling the resolution of a mesh

Let's imagine you have a mesh and you need to double the number of control points without changing the shape of the mesh. Editing the resolution attributes in the property window/spec tab does this. For example, you can double the value of the 'Count V' field.

Edit the 'Count U/V fields to control the resolution of the mesh

Note: if the resolution is increased, the shape of the mesh is not changed. Decreasing the resolution usually changes the shape.

Opening/closing meshes

You can close an open mesh and open any closed mesh by clicking the 'Close' tool. The tool opens the mesh from its 'start curve'.

If you have a number of meshes and you want to make them all closed, select the 'Close' pop-up menu item associated with the 'Close' tool.

Note: you can also use the 'Break' tool to open a closed mesh from an arbitrary position.
 

Inverting meshes

The 'Invert' tool allows you to change the mesh so that it will pass through its original control points. Just click the tool button to apply the tool.

For example: if you import a polygonal mesh and want to convert it to a smooth cubic mesh, inverting the converted mesh compensates for the 'shrinking' effect of degree elevation by smoothing.
 

Editing the current selection

The arrow keys (Up, Down, Left, Right) can be used for changing handle selection (curves/control points) in the 'Edit' mode. The 'Left' and 'Right' arrow keys can be used to move the current selection in the 'u' parameter direction. Correspondingly, the 'Up' and 'Down' keys allow you to move the selection in the 'v' direction. This is very handy when modifying complex meshes.
 
For example, to expand the current selection in the 'U' direction, hold down the Shift key and hit the 'Left' and/or 'Right' keys to select more curves.
 

Converting other objects to NURBS mesh form

The To Nurbs tool converts geometric objects to NURBS form by approximating them with a set of NURBS meshes. For example, to convert an analytic cube and sphere to meshes:

1. Multi select the cube and the sphere.

2. Activate the To Nurbs tool from the 'NURBS' tool tab.

3. Specify the approximation density values 'Density U' and Density V' in the tool control bar. The default values U=1, V=1 define the minimum density. Set U=2, V=2. These values are sufficient for most cases. Click Accept to finish the conversion.