by Reinhard Siegel
There are many situations in the modeling of the geometry of hulls and superstructures, where a basis surface requires some rounding. Being small in proportion there are no sophisticated freeform requirements, except to look nice and touch adjacent surfaces smoothly.
This article shows methods how to employ the surface type B-spline Lofted Surface for a fair attachment of classic hull keels, to round the bow or the stern of a hull, to remove sharp waterline entries. Its aim is to explain the pros and cons of various procedures, so in the end you can decide which method will suit your needs best.
cp: control point (support point)
mc: master curve = support curve
In the following the terms used for point, curve and surface types are those of MultiSurf. This may serve the understanding and traceability.
The tangential property of B-spline Curves
It is a fundamental property of the B-spline Curve, that it always starts tangent to the first segment and always ends tangent to the last segment of the polyline through its cps. This characteristic offers useful applications.
We can use the tangential property to join two B-spline Curves smoothly. All what is required is, that the neigbours to the common point are in line. This relationship is easily hardwired by a simple bead-on-line construction. (Model tangent_BCurve_joint.ms2).
Also the link between a B-spline Curve and a curve of some other type is not difficult. For example, we can make the bow mc run smoothly into the fairbody curve by this construction: put a bead on the curve where the B-spline should touch, create the tangent in this point (line from bead to a Tangent Point based on it) and use a bead on this line as last but one control point for the B-spline Curve. (Model tangent_bow_mc.ms2).
If the last cp of a B-spline Curve is a point on a surface (a magnet), a B-spline Curve will touch the surface in this point if its last but one cp lies on the tangent plane at the magnet. The tangent plane is defined by the normal vector at that point. (Model BCurve_tangent_at_magnet.ms2).
If the last but one cp of the B-spline Curve is a Projected Point on the tangent plane, then this tangent continuity is hardwired.
A Snake is a curve constraint to lie on a surface, and a Ring is a point constraint to lie on a snake, thus a Ring is a point on the surface. A Tangent Point, which is based on a ring, will lie in the tangent plane at the ring. Thus, in order to make a B-spline Curve touching a snake at a ring its cp next to end must be a Tangent Point or a bead on the line of tangency. (Model BCurve_tangent_at_ring.ms2).
As its name says, in a B-spline Lofted Surface the lofting curve is a B-spline Curve. In order to make a B-spline Lofted Surface touch an other surface one might have the opinion to use as its supports two snakes on that surface. The points on the first one defines the start, the points on the second one should ensure tangent continuity of all the lofting B-spline Curves and thus of the lofted surface itself. If both snakes are close together, the result might look not bad, but this construction does not guarantee the feature strived for. (Model non_tangent_BLoftSurf.ms2).
However, it is possible to create tangency by the following construction:
We can use the foregoing explained method to model true tangential roundings, bow and stern roundings, tangential attachment of a keel, a bulbous bow and so on.
Attachment canoe body – keel
The model attached_keel.ms2 shows canoe body and keel of a traditional sailing yacht hull. The keel is a B-spline Lofted Surface defined by 6 mcs, including the turn of the garboards. The keel surface starts along snake1 on the canoe body surface, and the tangency continuity is build in as in the previous example (Procedural Curve based on snake2 and the tangent plane at the momentary location of a ring on snake1).
If both snake1 and snake2 start on centerplane and the tangent plane is not vertical here, then the start of the Procedural Curve will be a little bit away from the centerplane. This would result in a small gap along the leading edge of the B-spline Lofted Surface.
Intersect the Procedural Curve with the centerplane, create a SubCurve with this intersection bead and use the SubCurve as 2nd mc for the B-spline Lofted Surface. Although by this quick fix a point on the SubCurve for a given value of t will not exactly lie on the tangent plane at a point of equal t on snake1, the deviation is very small and neglectable with regard to the accuracy one can achieve in the workshop.
In a logical sense it would be better to extend snake2 a little bit forward, until the Procedural Curve starts exactly on the centerplane. This will maintain the direct correspondence between the points on the 1st mc (snake1) and the 2nd mc (Procedural Curve) of the B-spline Lofted Surface.
To extend snake2 a little bit forward put a ring is on it, which in turn supports a SubSnake. The ring for this SubSnake must be positioned to make the Procedural Curve just start at the centerplane; a small negative t-value for the ring will by typical. This search for the ring position can be done manually, but when the model is changed, it must be repeated.
However, with a Solve Set entity MultiSurf will search the correct ring location automatically. This is implemented in model attached_keel.ms2. The Solve Set ss1 contains ring1 on snake2 as the free point; bead1 is at t=0 on the Procedural Curve, and plane1 is the centerplane. The Solve Set moves ring1 along snake2 until the distance between bead1 and plane1 is smaller than the tolerance. When the model is changed the Solve Set enforces the constraint relationship.
Attachment hull – bulbous bow
Another example of the same method is model bow_bulb.ms2.
A variation of the previously shown method is demonstrated in the model classic-1.ms2. Here the bow mc of the basis hull surface has all its cps on the centerplane (bow ghost line), thus the end of the waterlines are sharp. In order to round the entry of the keel waterlines and as well as the corner between leading edge and bottom of the keel a rounding surface is added. This is a B-spline Lofted Surface on 3 supports:
Note, that the tangent plane is actually a surface entity. This is necessary, because the projection direction is not normal to the tangent plane, but normal to the centerplane of the boat hull. With a 2 Pt Plane the direction of projection cannot be different to the normal vector of the plane itself.
Bow rounding – nearly tangential
Suppose we have a basis hull surface where the bow mc is the half siding of the stem. Then we will get a bow rounding B-spline Lofted Surface supported by
This cheap construction is demonstrated in model sailboat_bowround.ms2. As long as the hull surface is more or less flat lengthwise, there is nearly tangent continuity between the two surfaces. Sailboat hulls tend in this direction, but with powerboats and their full topside waterlines a break in slope will be noticeable.
It is very convenient, if the bow mc defines the half siding of the stem. Then the shape of the stem in profile view is determined directly by the x an z coordinates of the cps, the y coordinates of the cps define the bluntness of the stem. If the bow mc is the stem ghost curve on the centerplane, then there is just an indirect relation. If the actual stem in profile is fine, but its face is too small or wide, the shape of the foreship must be edited to make the waterlines wider or finer. In the author´s opinion a direct control of the prominent curves of a design is very important.
However, this simple bowround construction fails when used with a hull surface that is visibly curved in the fore-and-aft direction. This is shown in model bad_bowround.ms2 for the rounding of a bulwark basis surface. There is a considerable deviation between the tangent line to the top edge of the bulwark at point r1 and the line r1 to p11, to which the top edge of the bow round surface is tangent.
Bow rounding – true tangential
However, it is possible to create tangency by the following construction:
Along this lines the slope discontinuity of the imperfect bow round is fixed in the model good_bow-round.ms2.
This approach is also shown for the sailboat hull in the model angential_bowround.ms2.
Bowround by Blend Surface
Among the MultiSurf Examples there is the model Bowround2.ms2 where the rounding of a bow is accomplished by the surface type Blend Surface. For the sailboat hull above this method is pointed out in the model bowround_blend.ms2:
The Blend Surface property “Type” controls the smoothness of the join between the Blend Surface and the basis surfaces: Type = 1: slope continuity; Type = 2: curvature continuity.
However, as the comparison between the tangential bow rounding and those with the Blend Surface type 1 or type 2 reveals, the shape of the waterlines is not in favour for the latter. The Blend Surface bowround with curvature continuity shows even blunt waterlines (due to the flatness of the stem face).
The construction of a stern rounding follows the same principle as described for the true tangential bow round. (Model stern_round.ms2).
Work-around in case the projection fails
The construction of bow and stern rounding by a touching B-spline Lofted Surface is based on the procedural curve, which repeats these 3 steps for all positions of a ring on the snake which defines the start of the rounding:
In case there is a failure due to a too small angle between the line of the projection and the surface the magnet is to lie on, then this is the work-around:
The result will be a stern round, which is open at its bottom end. The longer the SubSnake and the SubCurve the smaller the hole. Close the hole by a Tangent Boundary Surface, a kind of surface, which allows to impose tangency and curvature continuity conditions along its boundary edges.
So far to the modeling of a B-spline Lofted Surface to make it touch an other surface.