When it comes to styling a surface, there is nothing more important to understand than the concept of Conic Curves and Conic Surface Laws.  You might ask, “What is so important about Conics when styling surfaces?”  The answer lies in the esoteric.  When you are looking at products on the shelf or in the showroom, there is something at work that you may not even realize as a consumer…Style  As a society of consumers, we are impacted by styling first, functionality second, and most providers of goods and services have realized this long ago.

First, let’s take a look at the theory behind Conics.  To start, we need to look at a simple radial fillet.  When fillets are created, tangency can be implied but there is an immediate change in radius, as represented below:

 

Notice that the tangency symbols displayed above occur at the ends of the lines where the fillet begins.  At this point, the curvature immediately changes from infinity (the radius of a line) to whatever radius value I choose for the fillet.  This means there is no smooth transition from line to fillet.  This poses a problem to stylists when they are considering reflections and “product presence”.  Any product that is allowed to use fillets, rather than conics, will usually be found in industrial applications, hidden away in rooms.  If it is to be seen, felt or heard by a consumer, a conic is usually in order.  Below is a sample of a reflection and how it is affected adversely by an instant change in radius:

 

The controlling factors for Conics are the surrounding surfaces and a conic parameter.  The surrounding surfaces contribute their tangencies and curvatures at their very extremity, in order to begin a smooth transition that will eliminate the visual edges that you saw in the previous example.  The Conic Parameter can be seen as a weighting of the fillet , which affects the resultant shape.  If you add weight to the fillet, it hyper-extends the curvature and tangency, and if you remove weight the fillet relaxes.  The hyper-extension of the fillet creates a ‘hyperbola’ and the relaxation of the fillet creates an ‘ellipse’.  In the middle of the road lies a solution called a parabola.  Take a look at the planar curve examples below:

 

Elliptical Conic

Parabolic Conic

Hyperbolic Conic

As for the three-dimensional results, pay special attention to the transition from the flat to curved surface and how the highly weighted Hyperbolic surface differs from the slightly weighted Elliptical Surface:

 

Elliptical Conic 1

Parabolic Conic 1

Hyperbolic Conic 1

This demonstrates that as the weighting of the Conic increases, the curvature and tangency extension increases.  Where the two flat surfaces would intersect is called the ‘Tangent Intercept’ curve, and represents the Apex or tip of the Conic in 3D space.

As for the use of Conic Laws, picture yourself being able to change the Conic parameter from Elliptical to Parabolic to Hyperbolic at will.  I will outline a challenge for you, along with some hints for successful completion.

The Parameters for Conics lie in the following ranges:

.5< Ellipse >.75          Parabola = .75                        .75< Hyperbola >1

Keep in mind that the law should be distributed along a line that is exactly the same length as the Spine for the surface.  This will simplify the application of the law against the support surfaces and allow you to work with literal distance values rather than ratios.

A Graphed Conic Law may look like this:

 

The Intermediate step in V4 looks like this:

 

And finally, the surface result looks like this:

 

Shown in V5 for clarity

You can see that the reflect line is more obvious at the elliptical end of the scale, and that the tip of the hyperbolic surface is a bit extreme.  For this reason, a lot of styling lies in the range just above and below the parabolic.  If you have problems conquering this concept, I can be contacted via email at mailto:catiadr@gmail.com

HAPPY SURFING!