# Splines and Surfaces #1: The history and theory as applied to CAD.

 Blog entry by SirIrb posted 08-13-2015 11:09 AM 2745 reads 1 time favorited 0 comments
 no previous part Part 1 of Splines and Surfaces series no next part

Placed here per request. This was written by a Unigraphics (NX) instructor named Ron Gardner. You will be a bit lost if you are not familiar with the program. Also the pictures didnt transition to the blog.
Steve

Understanding Splines and Surfaces

From the Main Menu expand Insert- Curve and look at the options.

Helix, Law Curve, Curve on Surface, Spline, Studio Spline, Fit Spline and Text are all splines. However, in NX there is only one mathematical recipe; one algorithm, for creating splines and while all of these buttons are different ways to tailor a spline the resulting curve is always fundamentally the same

NX, like most other CAD programs uses the NURBS algorithm.

Non Uniform
The points defining the spline can be randomly spaced.

Rational
A math definition meaning it deals with real numbers.

Basis-Splines
Basis refers to the weighting of the spline controls and is another mathematical concept.

The purpose of this paper is to demonstrate that although splines in a CAD environment are all mathematical the designer does not need to know the math to make efficient use of them. Far more important is an understanding of what the terms mean and how the various options control what the math is doing behind the scene. You can have NX open and click along as I go through the various concepts but it is not necessary as ample illustrations are included with the text.
NX Splines are true 3D curves. They can duplicate lines and conic sections but are not restricted to a single plane. The term Spline was first coined and used in conjunction with the lofting of ships and then later in aircraft and autos.

Recalling my own experience I once worked in a sail loft making yacht sails. One of my tasks was to “loft” full size sail layouts. The process involved measuring and locating defining points from a scale drawing onto the loft floor in full size. Once these defining points were located long flexible strips of wood or aluminum were placed to connect them. The strips (correctly called Splines) were held in place with lead weights called Ducks. The objective was to achieve a smooth or “fair” curve connecting all the defining points. The natural stiffness of the wood or aluminum spline made the least tension transition from point to point.

The curve was fair when the lofter had all his ‘ducks in a row’.

The challenge presented to early computer graphic developers and mathematicians was to duplicate this process of creating smooth curves in the virtual environment.

So moving to NX:

Discussion 1: Points and Poles

Change the view to Top and from the Curve toolbar click Studio Spline.

The Type drop-down menu has two choices Through Points and By Poles. The default Degree is 5. The Degree is an important concept that we will cover later.

Choose Through Points for the Type. Click a few times in the graphics window. As you click a spline is previewed connecting all the points you are selecting. Note that the spline appears to touch all the points.

Do not click OK yet.

Click and drag any one of the points you just made and note that you can reshape or edit the spline dynamically. Click Apply to complete the Through Points Studio Spline.

Change the Type to By Poles and click at least six locations in the graphics window. This time the spline begins and ends exactly on the first and last location picked but does not touch at any intermediate location.

The Poles, the locations you picked, are connected by a temporary linear framework called a Control Polygon. Click and drag on any pole and you edit the spline dynamically.

Click and drag the second Pole in from either end (the first interior pole) and, by observation, note that the spline remains tangent to the Control Polygon line connecting the second pole and the end pole. OK to complete the spline.

Double click the first, Through Points spline to edit it. Change the type to By Poles. The spline display shows a control polygon and poles.

Regardless of whether you choose Though Points or By Poles to initially create the spline the system always calculates the locations of the poles first. From the Poles it then passes the spline through the Defining Points if required.

A spline created By Poles has no defining points. See the Alert message in the illustration.
When you edit a Through Points spline by changing the type to By Poles the original defining points are deleted.

NX8 allows you to toggle between Through Points and By Poles when editing but the defining points are always deleted and redefined by the system when you do so.

Summarizing up to this point.

Through Points and By Poles are two ways to define a spline. Poles are the vertices of the Control Polygon and Defining Points are the points you pick for the spline to pass through. A spline is always tangent to the first and last leg of the control polygon.

Use a spline Through Points if you need the curve to pass through specific targets. Use a By Poles spline if you are more interested in a visually smooth transition.

So what’s a Knot and what is meant by CO, C1 and C2?
Activity: Create Studio Splines using Points

Activity: Create a Studio Spline using Poles

Discussion 2: Segmentation and Degree

The Studio Spline is the preferred tool for most spline construction however there is another, older tool simply called Spline. I will use this “old” spline to demonstrate segmentation and degree simply because; being old it is very manual. Some concepts are easier to illustrate without the dynamic interface that makes the Studio Spline so convenient.

Click Spline. Observe the first dialog and note the choices: By Poles, Through Points, Fit and Perpendicular to Planes. The choices, Through Points and By Poles are exactly the same as in the Studio Spline. I choose Through Points.

In the second dialog note the default is Multiple Segment and the Curve Degree is 3. Accept the defaults, OK. The next dialog is the choices for specifying the points to define the spline, i.e., the Defining Points.
Click OK in all the remaining dialogs and the spline is created.

Choose Point Constructor to get to the Point dialog and then click six locations, left to right, in the graphics window.

The resulting curve demonstrates non-uniform spacing and the fair, or smooth, transition between any numbers of defining points. Also note that the curve appears to pass through the defining points.

It is rather easy to visualize how the computer spline is behaving like the wooden lofting spline held in place by ducks.

Recall that this is a Multi-Segment Degree 3 curve. Understanding the relationship between Segments and Degree is the first step in understanding both Splines and Surfaces.

Going back to the “old” spline dialog, click Single Segment and note that the Degree field is grayed out. In the next dialog choose Point Constructor and click two locations in the graphics window.

OK all the dialogs and the resulting curve is a straight spline connecting the two points.

Click the old Spline tool again and note the value in the degree field is 1.The default Curve Type reverts to Multiple Segments but the dialog memory recalls the degree of the spline just created. Two points were picked the resultant degree was 1.

Again select Single Segment, continue through the dialogs and pick three points in the graphics window. OK until the spline is created.

Reopen the old Spline tool and the value in the Degree field is 2.

Make another Single Segment spline this time picking four defining points and when you reopen the spline dialog the degree will now show as 3.

From this you can infer that the degree for a single segment spline is one less than the defining points selected. Conversely, and this is probably the more important observation, you need one more point than the desired degree to define a spline segment.

Now hold that thought.

Degree
Let’s graph a curve for some numbers using the equation Y=X.

The result is a straight line. It is straight because the exponent of the equation is 1.

The highest exponent in the equation is the degree of the equation.

When you set the degree you are telling NX what to use for the exponent in the equations it creates to draw the spline. A degree 1 equation is linear and the transition the curve makes from point to point will be linear

Graph the same X values but this time with the equation exponent changed to 2.

The exponent of the equation, the degree, is 2 so the plot is no longer linear. It is now a conic section; in this case a parabola. Curves plotted from degree 2 equations will approcimate conic transitions from defining point to defining point.

Plot the same X values again but with the equation changed to Y=X³ and the curve, now a cubic (Degree 3) has a reversal of curvature and negative values.

In the simplest of terms; the degree controls the ability of the curve to bend.

So there are two concepts that we need to keep in mind and expand on. First, it takes one more point to define a spline segment than the spline degree and second, the degree controls the ability of the spline to bend.

Take this understanding back to NX. This time select Studio Spline. The Studio Spline offers many options including associativity that are not found in the old spline tool. Note also that the default Degree is 5.

Change the degree to 1. Click to select six screen points in the graphics window. OK to complete the Studio Spline.

How many curves are there? One.

Move your cursor over any part of it and you will see the entire curve highlight. It is one spline; the transition between each defining point is linear (straight) and it has corners.

The degree is 1 and, recall that it takes one more point than the degree to define a segment, so between every two points is a spline segment that is linear. Six points at Degree 1 required five segments to connect all the points. Each segment represents a unique math expression and all the segments are combined into a single spline.

In math terms a NURBS spline is a piece-wise curve: a single object made up of multiple expressions.

Make another studio Spline. Click six points in a configuration similar to the previous spline but this time set the degree to 2.

The transition is no longer linear.

Once again; make another spline as before but with the degree set to 3.

Check the information that NX stores on each spline. From the Main menu select Information-Spline. Accept the defaults and click OK. In the graphics window click on each of the splines, top to bottom, OK the Spline Analysis dialog and the Information window displays.

Recall that spline #1 is the first spline. The Information window confirms what we already know: the degree is 1; 6 points were selected so there are 6 poles and because it is degree 1 there are 5 segments. Note also that there are 4 C0 Knots.

Discussion 3: Continuity and Knots

When we set the degree to one and picked the six points we created one spline that had linear segments between each defining point.
Each segment is a separate math expression for the portion of the spline between the defining points.

The endpoints of each segment are called knots.

The information window is displaying the continuity condition across the interior Knots (endpoints of splines are also Knot points but for interior continuity, they are not considered.)

In the case of a Degree 1 spline the defining points, the poles and the knot points are exactly the same. As you will see this is the exception rather than the rule.

Look at the other entries in the information window

All three splines use an equivalent six points. As the degree increases the number of segments decreases. This makes sense because we already know that one point more than the degree is required for each segment. Six points at Degree 1= 5 segments. Six points at Degree 2= 4 segments. Six points at Degree 3=3 segments.

Set the degree to 5, the highest degree possible with only 6 points and the result is a single segment spline with 0 interior knots.

From the information window you see that when the degree is 1, the transition (continuity) across all the segments knots is C0. When the degree is 2, the transition across knots is C1 and for degree 3, C2.

The “C” number refers to the smoothness of the transition between segments across the knots.

Continuity

The C numbers are used to describe transitions across knots internal to an object but the same concepts apply to the transition across edges where two surfaces or faces meet. When describing the transition across edges the terms used are G0, G1, G2 and G3.

Continuity, or smoothness, is what is being described by the C and G numbers.

For example: take two objects meeting at a point. By observation we know the two are not Tangent. One way to prove they are not Tangent is to use Measure Angle. Click on each object.

The objects are tangent only if the result is 0 or 180 degrees
Two objects (surfaces, lines, faces etc.) touching but not Tangent is called G0. If it is a spline we are referring to, the “not tangent” transition across knot points is called C0.

Look at another construction, this time a horizontal line and an arc. The arc center is perpendicular to the endpoint of the line and its distance from the line is equal to the arc radius. Measure Angle between the line and the arc shows 180°confirming that they are Tangent.
When two objects (faces, surfaces etc.) are tangent, the transition across the joint is called G1.

To be tangent G1 objects do not have to be G0 they only have to have the same slope at the point where the measurement is taken.

The transition across Knot points in a multi-segment, degree 2 spine is tangent, C1.

Continuing with the line and arc example; the radius, measured anywhere along the arc will always be the same.

Similarly the radius anywhere along the straight line will be the same (∞).

But note that the line and the arc touch at a common end point and the radius, at that point, is different depending upon which object, the line or the arc, is being referenced.

The line and the arc touch (they are G0) and the angle between them is 0° or 180°, (therefore they are also tangent, G1), but they do not have the same Curvature (G2).

To be G2, objects must first be G0 and G1.

G2, called Curvature Continuity, is when two objects touch, are tangent and also have the same curvature at the point where they touch.

Internal to a spline, if the transition between segments across the knots is tangent and the curvature is also the same across the knots, the spline is C2.

All degree 3 or higher NX splines are curvature continuous (C2) throughout.
In the physical world of lofting, defining points are connected to make smooth curves by ducks and splines. The ducks hold the wooden spline in contact with all the points at the same time making a perfect physical analogy for the CAD single segment spine.

When all the points can be connected in one pass there are no segments and no knots so the transition is by definition perfectly smooth.
When the curve has to be “pieced together” using multiple expressions (recall that a multi-segment spline is a piece-wise NURB curve) there are segments, the ends of the segments are knots and the transition across the knots can be C0, C1 or C2. In other words, if a spline is multi-segment the knots allow different internal continuity conditions.

Curve Fit

On the drafting board a French curve is used for the same task as the ducks and splines on the lofting floor.

The French curve is “fitted” to a portion of the point set, matching as many points as possible, and a segment of the curve is drawn.

The French curve is then slid along the point set, overlapping the first drawn segment to establish tangency, and aligned with new points. Another segment is drawn connecting as many new points as possible

After drawing the second segment the French curve is slid further along the point set to draw additional segments connecting more points but always overlapping the previous segment.

The ‘draw, then move and overlap’, process is repeated until all the points have been connected by smooth curve.

The number of points overlapped determines how smooth the curve is.

NX uses a similar technique to create multi-segment splines. You select the points you want to connect and set the degree. NX defines segments by collecting one more point than the degree. It continues making segments by selecting points enough to satisfy the degree (plus one) overlapping the previous segment to establish continuity. The number of points overlapped determines the smoothness of the completed curve.

If no points overlap (Degree 1) there are corners and the transition from segment to segment across the knots is linear C0.

If two points overlap the transition is Tangent C1. Three overlapping points give C2, curvature continuity.

NOTE: I am using the term ‘overlap’ loosely to describe the way the system includes the poles toward the end of one segment as first poles in the next segment.

Checking for Smoothness

The most common tool for visually checking the smoothness of curves is the Curvature Comb. It is accessed from the Main toolbar-Analysis-Curve Analysis or the Analysis toolbar.

Looking again at the three Studio Splines; hide the Degree 1 and Degree 3 examples.
From the Analysis toolbar select Curve Analysis and click the Degree 2 spline. The Curve Analysis options will be discussed in depth later. For now the default settings will suffice.

The display of lines radiating perpendicular to the spline is the Combs and each Needle in the comb represents the Curvature at that point on the spline.

The display of curvature needles is inversely proportional to the curvature so a shorter needle indicates a tighter radius than a longer needle. The outer ends of the needles have a connecting curve to help visualize the smoothness of the spline. You can think of this connecting curve as an exaggeration of the spline’s wobbles to aid evaluation.

The Curve Analysis tool shows Inflections, places where the curvature reverses. Inflections are not always as easy to spot as in this example. Unintentional inflections have a nasty effect on NC machining among other faults.

This Degree 2 spline used six defining points. Given 6 points minus degree 2 we can expect that there are four segments and one knot at the end of each segment. We can verify that by checking Information-Spline again or we can display points at the knot locations.

From the main toolbar click Insert-Datums/Points- Point Set. In the Point Set dialog set the Type to Spline Points, the Sub-type to Knot Points and click the spline in the graphics window.

Five knots display confirming our expectations. Note that the knots appear to be at locations that have radical changes in curvature.

Double click the Curve Analysis display to return to the dialog. Change the number of needles to some much higher number.

You can now see that there is a radical change in curvature at every internal knot. The spline is Tangent (C1) across all the knots but the Curvature is different depending on the segment being referenced.

Use Show to display the Degree 3 spline and hide the Degree 2 spline. Edit the comb to show more needles and make points at the knots as with the previous example.

Six defining points minus Degree 3 equals three segments. Note that the curvature comb display has equal length needles on either side of the interior knots and the curvature flows more smoothly. The Curvature across the knots is C2 so the transition is smoother but closer observation will show that despite being C2 the curvature change can still be rather sharp.

Anytime there are Knots there is the opportunity for the curvature to change radically. A single segment curve has no knots therefore the curvature must be continuous and smooth without abrupt change.
Double click the Degree 3 spline to edit it. Change the degree to 5. Six defining points minus degree 5 equal one segment and no interior knots.
There are two obvious questions that come to mind.

The first is: Why do I care about this C and G stuff? Without the comb display the splines and surfaces all look pretty much the same.

And second: Why use a multi segment spline at all? Single segment splines are smoother.

Let’s take the “why do I care” question first.

The answer, quite simply is; maybe you don’t care.

Determining what is ‘good enough’ on the CAD model is very subjective and is dependent upon the product. A structural part invisible inside a machine does not need the same quality as an exterior automobile panel. Many factors play in determining what is ‘good enough’. Does the part have texture? What material is it? Does it get polished or go through secondary operations after casting or machining?

However: there is no down-side with having the highest quality. Poor quality invites problems such as CNC machining faults and difficulty applying Blend and Shell features to the model. In the worst cases a model may not sew to produce a useable solid or it will fail to update when minor edits are attempted.

Quality, in this context, means that a surface is smooth without unwanted inflections or changes in curvature and also that surfaces meet with minimum deviations (G0) and have appropriate continuity (G1-G3) for the desired application.

Since most freeform tools in NX rely on Curves for their definition, high quality curves are essential to design success. If those curves are splines then the choice of Degree and Segmentation play an important role; as does continuity where objects meet.

Continuity across edges, G0, G1, G2 and G3

This will be easiest to see if we take two single segment splines (perfectly smooth) and look at the transition at their junction.
From the curvature comb you can observe that the curves are not Tangent. Their endpoints do touch so they are G0. A simple extrude of the two curves produces sheet bodies. Using the Face Analysis-Reflection tool to display zebra stripes on the surfaces you can see that the transition across the edge is sharp. There is a noticeable break as the stripes cross the edge.

Using the same curves but making one G1 (tangent) to the other the combs are now parallel at the junction indicating the slope is the same at that point.

Observe the extruded surfaces.

The zebra stripes are continuous and flow across the joint but there is considerable distortion of the image. A tangent transition is smooth but it always focuses light at the edge. It can create the appearance of an unwanted ‘bone line’ in an aesthetic surface.

Make one curve G2 (curvature continuous) to the other.

With G2 continuity the stripes flow across the edge making a more visually acceptable transition. G2 is usually an aesthetic concern rather than structural.

G3 is the smoothest transition option. The objects touch, are tangent and are curvature continuous. In addition the rate of change, the acceleration, is the same for both objects across the transition.

G3 produces a visually seamless transition.

So the more complete answer to the “Why do I care” question is: A designer needs to have a quality level that, at minimum, makes a stable model that will sew, blend, shell and update. Additionally, and this is dependent on the end use, the designer needs to choose transitions that give the desired visual affect.
• G0 (sharp transitions and edge blends) appropriate to hidden parts, industrial products and similar products where aesthetics are not important.
• G1 for aerospace, flow products and many medical applications.
• G2 All visible automotive surfaces and many consumer products.
• G3. Special conditions where exceptional transition quality is required such as across the mid-plane of an auto roof or deck lid.

In very general terms:

Having addressed the ‘Why do I care question’ let’s move onto the next.

Why use a multi segment spline at all? Single segment curves are smoother.

This is actually a much easier question. Before jumping into it though I want to reiterate that all of the concepts and terms that apply to splines also apply to surfaces. Everything you know about splines applies to surfaces. Not every surface depends on a spline but the principles are exactly the same and if you can make good splines you can make good surfaces.
Defining Points of a spline (the points you pick for the spline to pass through) are equivalent to the Section Strings (the curves you pick for the surface to pass through) of a surface.

A spline has a Degree. From the beginning to the end of a spline along its arc length is called the U parameter or, more loosely, the U direction. A surface has both a U and a V direction. There is a degree in both U and V.

All splines have poles and a control polygon. All surfaces have poles and a control polygon.

A spline is Multi-Segment if it has more poles than its degree plus one. A surface is Multi-Patch in U, V or both if there are more poles than the degree plus one in either direction.

A Multi-Segment spline has knots. A Multi-Patch surface has Knots.

I said at the beginning that regardless of which tool you choose, behind the scenes, NX is always using the same math to define the curve. In the old Spline tool there is a button to choose Single Segment. In the Studio spline there is no button, the segmentation is automatic. If you choose one point or pole more than the user defined degree the resulting spline is single segment otherwise it is multi-segment.

Here is an experiment.
Click Spline (the “old” spline) and choose Throiugh Points. In the next dialog choose Single Segment , Ok and then Point Constructor. Change the view to Top.
The challenge is to make a horizontal spline by clicking 18 or more defining points, left to right across the graphics window. NO CHEATING. Just align and click each point by eye!
Click, click click…………

………. …..click, click, ………………click, click,( whew) and click.

OK; OK; OK.

Hmmmm; not exactly what I wanted.

The degree is one less than the number of points you picked. In this case I picked 18 points, got a degree 17. High degree splines are extraordinarily difficult to manipulate. NX can create splines up to degree 25 but the highest degree practical for modeling applications is 7 and that is rarely used.

Just to confirm that this behavior is not some aberration associated with the old spline tool, do the same exercise with Studio Spline.
Instead of a button for single segment, with the Studio Spline you pick one point more than the user-defined degree.
From the Curve toolbar click Studio Spline, change the Degree to 17 and start picking 17 points.

Click, click…The preview is looking good…click, click…

Click, click,……………………OK

Not so good. The point set is different from the first example but the result is the same; an unusable curve.

Remember the question?
Why use a multi segment spline at all? Single segment curves are smoother.

I said the answer was easy. If you need to hit more than 7 or 8 targets the single segment spline is too stiff and does not behave well. It is smooth but unpredictable.

Another set of Very General guidelines:
• If you need the curve to go through 5, 6 or less points the decision is easy. Set the Studio Spline degree to one less than the number of points and the result will be single-segment.
• For six to 8 points, try a single segment as above and see if the result behaves suitably.
• More than 8 points choose degree 3 or 5 and select all the points. The result will be multi-segment (C2). Use Analysis-Curve-Curve Options to determine if the quality meets requirements.
• If you have many points that the curve must pass through and you need the highest quality, the solution may be to use several single segment splines with G3 continuity between them rather than just one.

Discussion 4: Tolerance

All freeform is tolerant. That is to say, whenever you click a target to attach a spline or surface, the position and continuity are accurate within specific limits. The global settings for tolerance can be seen in the Preferences-Modeling drop-down menu under the General tab.

Additionally almost all NX8 dialogs have fields to allow specifying tolerance on-the-fly. Tolerances set in Preferences-Modeling are the default and affect all features in the part file except for those where the tolerances are changed in the dialog.

Distance Tolerance
This is the maximum distance whatever feature you are making is allowed to deviate from the specifying geometry. For example: A spline created through a set of points will not be further from any of the defining points than the value set in the Distance Tolerance.

Similarly, a surface will not deviate from its section curves more than the distance tolerance. Checking the deviation (Analysis-Deviation-Curve to Face), in the following illustration of a Studio Surface through four sections, highlights an important consideration.

Note that although the Distance Tolerance is .001 all of the distance measurements are zero to eleven decimal places. The system only uses the allowable tolerance if it needs it. Having a relatively loose tolerance does not mean that surfaces and curves will necessarily have significant deviations. The distance tolerance is the G0 (positional) tolerance between two objects that are supposed to touch.

The next example is a single arc being used to create two Studio Surfaces. The arc is swept along the dashed curve to the left to create the first Studio Surface. The same arc is swept along the dashed curve to the right to create a second Studio Surface.

A check of the deviation between the edge of either sheet to the arc this time shows that there is a difference.
Double clicking either of the Studio Surfaces and editing the G0 Tolerance field to .00001 and rechecking the deviation shows the following.

G0 Tolerance File Size
.001” (default)
.00001”
.000001”
So what is the lesson?

First, all freeform is tolerant but just because there is a generous tolerance does not mean that the resulting fit will be excessive.

Second, features can be edited to tighten fits if necessary on an individual basis.

Third: there’s no free lunch, tighter tolerances do increase file size. Observe the difference in the very simple part above when the G0 tolerance of the two surfaces is made tighter.

I often hear users say that NX default modeling tolerances are not tight enough. That may be so but I am reminded of the time when I was designing my house. The site was not flat so being a freeformin’ dude I tried to model it in 3-D by recreating the contour curves from the survey plan. Memory was not as great then as it is today and my system crashed repeatedly until I realized that modeling a plot of land at ±.001” made little sense. I changed the tolerance to a more reasonable ±0.1” and the surface developed almost instantly.

Whether or not the default tolerance is too loose or too tight depends on what you are trying to model. Tolerances that are fine for a large fiberglass tank may not be satisfactory for surgical implant.

Angle Tolerance

The Angle Tolerance is the difference between the normal vectors of the parent object and the equivalent normal vectors of the child.
Perhaps you are making a Through Curves feature from several section strings as in this VERY exaggerated example. The Angle tolerance is labled at the point of crossing for clarity but it is calculated at all equivalent parametric locations along the curve and the surface. Since it is the difference between the normal vectors, the Angle Tolerance also determines how close G1 (tangency) is modeled. The default Angle tolerance is ±.5°. That means that as long as two objects are within half of one degree NX will treat them as tangent.

When analyzing quality by evaluating G0, G1 and G2 it is important to note that Distance Tolerance is in length units (mm or inch) and Angle Tolerance is in degrees. Curvature however, is a scale rather than a unit measure. The curvature scale is 0-2. 0 means curvature continuous while values approaching 2 means that curvature continuity does not exist or is not valid. An analysis of curvature continuity to a planar or cylindrical surface, for example, will return a value of 2.

Surface Continuity
(With Shape Studio: Analysis-Shape-Surface Continuity) is one tool for evaluating surface intersection quality. In the next example we are looking at a Surface Continuity display of a sailboat keel.

With the check set to examine G0 at the edge to edge matching between two transition surfaces, the display shows a worst-case value of 0.00000 indicating there is no gap between the edges. Show Continuity Needles is checked but since the G0 distance is zero the length of the teeth is zero and no comb is displayed.

Zoom in a bit and change the Comparison Objects to G1. The display now shows the worst-case G1 condition to be .05772 degrees, well within the default .5° Angle Tolerance. The length of the needles in the comb display is the proportional deviation from perfect tangency; the longer the needle, the greater the deviation up to the maximum .05772 value. Needles are above and below the intersection indicating positive and negative values.

Change the Continuity Checking to G2 and note the displayed value is 2.0000 and there are needles displayed. Both indicate that G2 continuity does not exist.

That concludes this discussion of Understanding Splines.

-- Don't blame me, I voted for no one.