AliasStudio Concepts Keypoint curves > Overview

Curves

Describes curves as they appear in the AliasStudio interface.

CVs, hulls, and edit points

Explains the origin and use of the different curve features.

CVs

CVs (control vertices) control how the curve is "pulled" from a straight line between edit points. They are the most basic and important means for controlling the shape of a curve. Lines between consecutive CVs form the control hull.

You cannot add CVs to the interior of a curve: there is always a set number of CVs for each span. The number of CVs is equal to the degree of the curve plus one. So, for example, a degree 3 curve has four CVs per span.

AliasStudio draws CVs differently to let you tell the difference between the start and the end of a curve. The first CV (at the start point of the curve) is drawn as a box. The second CV is drawn as a small "U", to show the increasing U dimension from the start point. All other CVs are drawn as small X's.

Multiple spans

Longer and more complex curves require more than a single span curve. As you draw what appears to be a single long curve, AliasStudio is actually adding several curve spans together. The last CV of the previous curve span become the first CV of the next curve span, creating very smooth transitions between the curve segments.

Edit points

You can tell when a curve is made from multiple spans in several ways. One is to look for edit points on the curve. Edit points mark the connection point between two spans. AliasStudio draws edit points as small crosses.

Unlike the on-curve control points of Bezier curves (used in many 2D illustration programs), NURBS edit points are not usually used for editing curves. CVs control the shape of a NURBS curve, and edit points are just indicators of how many spans a curve has.

There are, however, a few tasks that use edit points:

If you want more control in a curve, you can insert an edit point to increase the number of spans in the curve and give you more CVs to work with.
You can also delete edit points to decrease the number of spans in a curve (and probably change the shape of the curve).
It is possible to move edit points to change the shape of a curve, but you should avoid doing this except for minor adjustments.

AliasStudio does not actually move the edit point itself, but instead moves the CVs to reshape the curve so the edit point is where you specified.

Hulls

As a curve gets more spans/edit points, you might lose track of the order of the CVs. To show the relationship between CVs, AliasStudio can draw lines between them. These lines are called hulls.

(AliasStudio also provides other feedback to show the order of CVs. For example, when you pick a CV, AliasStudio highlights its span within the curve.)

Moving edit points vs. moving CVs

Describes why moving CVs is preferable to moving edit points when reshaping a curve.

In theory, moving edit points would be an excellent way to edit a curve, since they lie on the curve itself. Unfortunately, it doesn't work out that way. This is because the shape of the curve determines the positions of edit points, not the other way around.

AliasStudio does allow you to move edit points by "reverse engineering" the curve from the edit point. When you move an edit point, the Move tool tries to find a curve which passes through the new edit point location. Because this process is time-consuming and has an infinite number of solutions, the tool must place constraints on how moving the edit point affects the curve.

Because of these constraints, you usually cannot make major changes well by moving edit points. Moving edit points is best for small scale reshaping.

Even though it is slightly less intuitive, the only way to reshape the curve with complete power is by moving CVs.

Multi-knots and CV multiplicity

Describes two ways of achieving sharp bends in NURBS geometry. While these features are supported to a large degree in AliasStudio, they can cause problems with certain tools and other software.

A multi-knot is multiple edit points at the same location in space.

CV multiplicity is multiple CVs at the same location in space.

Multi-knots are usually the result of curve or surface editing operations that require a sharp turn in a curve. CV multiplicity is created by manually placing adjacent CVs in the same location (using the Magnet tool).

Multi-knots and CVs with multiplicity are generally undesirable. Some tools (such as Rail Surface) cannot work with them, and many CAD packages will not accept models with multi-knots.

Multi-knots and CV Multiplicity achieve similar effects, even though they are different mathematically.

> Multi-knots and continuity

Multi-knots eliminate one level of automatic continuity for each extra edit point.

For example, a degree 3 curve normally has curvature continuity (G2) at edit points.

If you create a multi-knot of two edit points, you lose automatic curvature continuity, so you only have tangent continuity (G1) at the multi-knot.
If you create a multi-knot of three edit points, you lose both automatic curvature and automatic tangent continuity, so you only have positional continuity (G0) at the multi-knot.

Only the intrinsic continuity is lost. As with Bezier curves, clever placement of CVs can restore continuity.

AliasStudio only creates full multiplicity knots, i.e. knots which have a multiplicity equal to the degree of the curve.

Rational vs. non-rational geometry

Explains the differences and pros and cons of rational and non-rational geometry.

Non-rational geometry is a sum of polynomials. Rational geometry is a ratio of sums of polynomials. Rational geometry is considerably more complex mathematically. Therefore:

It may not be transferable to downstream CAD packages that can't deal with complex descriptions
It can be slower to manipulate when modeling, and slower to render.

The following tables lists the differences between the two types of geometry.

Nature	Pros	Cons
Non-rational	· More flexibility for transformations. · Faster.	· Sacrifices some precision for modeling flexibility.
Rational	· Precise geometry (that is, exact conics).	· Weighted CVs not supported by many CAD packages. · Weighted CVs harder to manipulate. · Creates multi-knots. · Slower to display and render.

This illustration shows two circles drawn with the two types of geometry.

The circle on the left is a non-rational curve with CVs that are all weighted equally. To have a non-rational curve, all weights must be 1.0.
The circle on the right is a rational curve with different weights applied to the CVs, and multi-knots.

You can see the difference in two ways:

If you attach a radius measurement to the circles, you will see that the non-rational circle is not a perfect circle (although it is pretty close): it has different radii depending on where you measure. The rational circle is a perfect circle.
Attach curve curvature combs to the circles. The curvature on the non-rational circle on the left varies. The curvature of the rational circle on the right is constant.

Constructing quality curves

Contains tips for constructing curves that will make building high quality surfaces easier.

To create quality surfaces you need quality curves. These guidelines will help you create good curves.

Simple curves

Use the simplest curves that can describe the shape you want. Simpler curves mean simple, faster rendering surfaces.

One effective method for achieving simple curves is:

Begin a curve by drawing a single span.
Move the CVs to achieve the shape you want.
If you can't achieve the shape, add an edit point to create more CVs.
Continue until you have the shape you need.

This iterative process ensures your curve only has as many spans as are absolutely necessary.

You can also use the Rebuild curve tool to simplify existing curves. The tool can simplify a curve while maintaining its shape within a tolerance you set.

Parameterization

It is often best to build curves with uniform parameterization, because it makes inserting edit points and detaching curves at exact locations easier.

When drawing Edit point curves with Uniform parameterization, the resulting CVs may be placed awkwardly. To fix this, move the CVs to prevent crossing hull lines.
Try to consistently use either Uniform or Chord length parameterization when drawing curves. If you mix and match curve styles, it could result in cross knot insertion when the curves are used to build a surface.

Intersections

Some surfacing tools require curves to intersect:

To draw intersecting curves, use curve snapping (hold down + , or click the curve snapping button to the right of the prompt line).
To make existing curves intersect:
Pick an edit point and use the Move tool with curve snapping.
Use the Object editor with curve snapping.

Planning for surfaces

When creating curve, plan ahead to the surfaces that you want. Try to have the same number of spans in all the construction curves for building a surface. A simple way to achieve this is to start with one curve, then duplicate it to create more construction curves.

When you create a surface from curves with different numbers of spans, the new surface will have an extra isoparametric curve corresponding to every extra edit point. This is known as cross knot insertion. It makes the new surface more difficult to edit and more complex.

Blend curves

Describes the features and concepts behind blend curves, which allow you to create curves by specifying constraints on their shape.

Palette tool: Curves > Blend curve toolbox

Blend curves provide higher-level, simpler methods for shaping and manipulating curves. They provide a level of abstraction on top of the actual geometry of the curve. Blend curves let you focus on what the curve needs to do, and have the system calculate the right curve to fulfill those requirements.

Blend curves are normal NURBS curves with more construction history: you can use all the normal curve tools on blend curves, and when you are not using blend curve tools, they look like any other curve.

Blend curves are controlled by blend points acting as constraints:

You create the curve by setting up the constraints, such as

what points in space the curve should pass through,
which surfaces it should be tangent to,
which existing curves the blend curve should intersect,
what direction it should be travelling at a certain point,

...and so on. AliasStudio draws the curve to satisfy the constraints, and automatically updates the curve when the constraints, or the objects the curve is constrained to, change.

Types of blend points

There are three main types of blend points.

Location: forces the curve to pass through the blend point's location in space.

This is the type you create when you first draw a blend curve.

Direction: forces the curve to pass through the blend point's location in space travelling in a certain world space direction.

There are two sub-types of direction:

Directed: you set an actual direction for the curve tangent. Use this type when the specific tangent direction at the point of the blend point is important.
Parallel: you set a line along which the curve passes (in either direction) at the blend point. This is easier to enforce and results in better curve continuity.
Geometry: forces the curve to pass through a point on a curve or surface and travel in a direction relative to that curve or surface.

The following table shows the icons used to represent the different constraints:

Type	Not attached	Attached to blend curve	Attached to regular curve
Location
Direction

Keypoint curves

Describes the concepts behind keypoint curves, which allow you to create CAD-like lines and arcs.

Overview

Keypoint curves retain more information than other curves. They remember relationships and constraints, and apply them when you edit the lines. You can also edit these special attributes in the Information Window.

For example, a keypoint arc has edit points and CVs just like a normal curve, but it also has a radius, sweep angle, and center point, all of which can be edited. During editing, the arc stays an arc: it will not lose its shape from keypoint editing.

When you combine keypoint curves into composite curves (for example, with the Line-arc tool), relationships between the individual lines and arcs are still maintained.

Keypoint curve tools create guidelines, which are very useful for aligning curves with each other as you draw.
Keypoint curves are especially useful for CAD and drafting applications. However, any part of your model requiring geometric accuracy or ease of editing will benefit from keypoint curves.

Most tools that work on normal curves also work on keypoint curves.