Developable surface

Basic examples

The developable surfaces which can be realized in three-dimensional space include:

• Cylinders and, more generally, the "generalized" cylinder; its cross-section may be any smooth curve
• Cones and, more generally, conical surfaces; away from the apex
• The oloid and the sphericon are members of a special family of solids that develop their entire surface when rolling down a flat plane.
• Planes (trivially); which may be viewed as a cylinder whose cross-section is a line.

Other properties

A surface defined by ${\displaystyle F(x,y,z)=0}$ is developable if^[3]: 93$${\displaystyle \left|{\begin{array}{llll}F_{xx}&F_{xy}&F_{xz}&F_{x}\\F_{yx}&F_{yy}&F_{yz}&F_{y}\\F_{zx}&F_{zy}&F_{zz}&F_{z}\\F_{x}&F_{y}&F_{z}&0\end{array}}\right|=0,\quad F_{x}=\partial F/\partial x,{\text{ etc }}}$$

Relation to ruled surfaces

Consider a ruled surface$${\displaystyle F(s,v)=c(s)+vd(s),\quad |d(s)|=1}$$with ${\textstyle s}$ the ruling parameter and ${\textstyle v}$ the coordinate along each ruling. It is developable iff the tangent plane does not twist as you move along a ruling, i.e. the tangent plane at ${\textstyle F(s,v)}$ is the same for all ${\textstyle v}$ with that fixed ${\textstyle s}$. In formulas, it says$${\displaystyle \operatorname {det} \left(c^{\prime }(s),d(s),d^{\prime }(s)\right)=0,\quad \forall s}$$When that holds, infinitesimally close rulings intersect, and they produce a space curve as their envelope curve. The curve is traced out by ${\textstyle c(s)+v_{0}(s)d(s)}$, where $${\displaystyle v_{0}(s)=-{\frac {\left\langle c^{\prime }(s),d^{\prime }(s)\right\rangle }{\left\langle d^{\prime }(s),d^{\prime }(s)\right\rangle }}}$$This curve is the edge of regression of the original surface, and the original surface is the tangent developable of the curve. tangent line to ${\textstyle r}$.

Generically, each developable surface has only one edge of regression. In this way, the theory of generic developable surfaces is the same as the theory of generic tangent developables.

Special developable surfaces have "degenerate" edges of regression, including a plane, a cylinder, a cone.

Tangent

Developable surfaces can be generated in infinite varieties with some general theorems.

As previously stated, the tangent developable surfaces are constructed by extending the tangent lines of a spatial curve. This is the envelope surface of the osculating planes of the curve. The curve itself is called the edge of regression of the developable surface.

The envelope surface of a differentiable 1-parameter family of planes is developable. In this way, any differentiable function of the type ${\displaystyle \mathbb {R} \to {\text{planes in }}\mathbb {R} ^{3}}$ produces a developable surface.

Polar

Polar developables are constructed as follows: Given a smooth space curve ${\displaystyle \gamma }$, construct the osculating circles at each point, and then the line passing each center of the circle perpendicular to the circle's plane. These lines are the polar lines. Equivalently, construct the normal plane at every point. The normal planes of two infinitesimally close points intersect at a line. These lines are the polar lines. The polar lines sweep out a developable surface, since it is the envelope surface of the normal planes of the curve.^[4]

For example, the polar developable of a helix is obtained by screwing a plane along an axis that is not within the plane.

For a generic space curve, the edge of regression of its polar developable is the curve of centers of osculating spheres of the curve. In many texts this is also called the focal curve or spherical evolute of the curve.

Given a space curve ${\displaystyle \gamma }$, and another space curve ${\displaystyle \mu }$, we say that ${\displaystyle \mu }$ is an evolute curve of ${\displaystyle \gamma }$ iff each tangent of ${\displaystyle \mu }$ intersects ${\displaystyle \gamma }$ at a right angle. This is analogous to the concept of the evolute in plane curves, although note that in the plane, there is only one evolute per curve, but in space there in general exists a whole 1-parameter family of evolutes per curve. All evolute curves of ${\displaystyle \gamma }$ are geodesics of the polar developable of ${\displaystyle \gamma }$.^[5]

Let ${\displaystyle \mu }$ be an evolute of ${\displaystyle \gamma }$, then ${\displaystyle \gamma }$ is in the tangent developable of ${\displaystyle \mu }$, and ${\displaystyle \gamma }$ is orthogonal to its ruling.

The polar developable is a cylinder iff the original curve ${\displaystyle \gamma }$ is a curve in a plane ${\displaystyle P}$ orthogonal to the cylinder's direction. In this case, it is obtained by translating the evolute curve along the perpendicular direction of ${\displaystyle P}$.

For example, let ${\displaystyle \gamma }$ be the involute of a circle. Regarded as a space curve, ${\displaystyle \gamma }$'s polar developable is the cylinder generating by moving the circle along its perpendicular direction. The evolutes of ${\displaystyle \gamma }$ are the helices on the cylinder, all passing the same cuspidal point where ${\displaystyle \gamma }$ touches the cylinder.

The polar developable is a cone iff the original curve is a curve on a sphere centered at the apex of the cone.

Normal

Given a curved surface, the normal lines along any line of curvature of the surface sweeps out a developable surface. In this way, a generic curved surface produces two 1-parameter families of developable surfaces, intersecting at right angles. These are the normal developables of the surface.

Take one of these two 1-parameter families. Each developable surface in the family has an edge of regression. These produce a whole 1-parameter family of edges of regression, and they sweep out a space surface. The other family sweeps out another space surface. These are the two focal surfaces of the original surface.

Geodesics

The geodesics of a plane are straight lines, so the geodesics of a developable surface can be obtained by first rolling it flat on the plane, drawing a straight line, then unrolling it back. This produces the 2-parameter family of all geodesics.

Examples

Spheres are not developable surfaces under any metric as they cannot be unrolled onto a plane.

The helicoid is a ruled surface – but unlike the ruled surfaces mentioned above, it is not a developable surface.

The hyperbolic paraboloid and the hyperboloid are slightly different doubly ruled surfaces – but unlike the ruled surfaces mentioned above, neither one is a developable surface.

Of non-developable surfaces

Many gridshells and tensile structures and similar constructions gain strength by using (any) doubly curved form.