What is the condition that the surface to be developable?
Two well-known theorems used in the proof are: (a) the tangent plane to a developable surface is the same at all points of a generator; (b) if the second curvature K of a surface vanishes identically, the surface is developable. such that the normals at P and P’ are parallel.
How do you find the Gaussian curvature of a surface?
Let κ1 and κ2 be the principal curvatures of a surface patch σ(u, v). The Gaussian curvature of σ is K = κ1κ2, and its mean curvature is H = 1 2 (κ1 + κ2). To compute K and H, we use the first and second fundamental forms of the surface: Edu2 + 2F dudv + Gdv2 and Ldu2 + 2Mdudv + Ndv2.
How do I know if my surface is developable?
Just orient the surface in space until you get a clear view straight down one isocurve. The surface edges at both ends of an isocurve on a developable surface will appear to be parallel, at that location, which means that they lie in the same plane.
What is the Gaussian curvature of a hyperbolic surface?
All sectional curvatures will have the same sign. If the principal curvatures have different signs: κ1κ2 < 0, then the Gaussian curvature is negative and the surface is said to have a hyperbolic or saddle point. At such points, the surface will be saddle shaped.
What is developable surface in differential geometry?
A developable surface is a special ruled surface which has the same tangent plane at all points along a generator [13,222,120,32,326,252,329]. Since surface normals are orthogonal to the tangent plane and the tangent plane along a generator is constant, all normal vectors along a generator are parallel.
What is a developable surface in surveying?
Developable surfaces are a special kind of ruled surfaces: they have a Gaussian f curvature equal to 0,1 and can be mapped onto the plane surface without distortion of curves: any curve from such a surface drawn onto the flat plane remains the same.
What does Gaussian curvature tell us?
Gaussian curvature is a curvature intrinsic to a two- dimensional surface, something you’d never expect a surface to have. A bug living inside a curve cannot tell if it is curved or not; all the bug can do is walk forward and backward, measuring distance.
What do you mean by a developable surface in map projection?
developable surface. [map projections] A geometric shape such as a cone, cylinder, or plane that can be flattened without being distorted. Many map projections are classified in terms of these shapes.
What is developable land?
Developable Land means land which is appropriately zoned, has access to all necessary utilities and has access to publicly dedicated streets.
What do you mean by developable surface in map projection?
How many types of developable surfaces are there?
There are three types of developable surfaces: cones, cylinders (including planes), and tangent surfaces formed by the tangents of a space curve, which is called the cuspidal edge, or the edge of regression. Cylinders do not contain singular points. The only singular point of a cone is its vertex.
What is Gaussian surface in physics?
The Gaussian surface is known as a closed surface in three-dimensional space such that the flux of a vector field is calculated. These vector fields can either be the gravitational field or the electric field or the magnetic field.
What is surface curvature?
To measure the curvature of a surface at a point, Euler, in 1760, looked at cross sections of the surface made by planes that contain the line perpendicular (or “normal”) to the surface at the point (see figure).
Which surface is not developable?
Spheres and other surfaces, that have compound curvature, cannot be unfolded or “developed” accurately without knowing something about the characteristics of the material (amount of stretch available and more.) Non-developable surfaces have compound curvature, that is, curvature in two directions, not just one.
What is the difference between developable and non developable surface?
Developable Surface : A developable surface is that which can be cut or unfold into a flat sheet or paper e.g., cylinder or cone. Non-developable Surface : A non-developable surface is that which cannot be cut or folded into flat sheet paper, e.g. globe.
What is a developable surface and what common shapes are used to represent them?
Developable Surfaces: A developable surface is a geometric surface on which the curved surface of the earth is projected; the end result being what we know as a map. Geometric forms that are commonly used as developable surfaces are planes, cylinders, cones, and mathematical surfaces.
What is the difference between developable and non-developable surface?
What is Gaussian curvature?
Gaussian curvature is the limiting difference between the circumference of a geodesic circle and a circle in the plane: Gaussian curvature is the limiting difference between the area of a geodesic disk and a disk in the plane: Gaussian curvature may be expressed with the Christoffel symbols:
What is the Gaussian curvature of embedded smooth surface in R3?
The Gaussian curvature of an embedded smooth surface in R3 is invariant under the local isometries. For example, the Gaussian curvature of a cylindrical tube is zero, the same as for the “unrolled” tube (which is flat).
What is the surface of a torus with negative Gaussian curvature?
From left to right: a surface of negative Gaussian curvature ( hyperboloid ), a surface of zero Gaussian curvature ( cylinder ), and a surface of positive Gaussian curvature ( sphere ). Some points on the torus have positive, some have negative, and some have zero Gaussian curvature.
How do you find the curvature lines of the developable surface?
In the remaining cases the developable surface is formed by the tangents to a certain space curve — the cuspidal edge (or edge of regression) of the developable surface. In this case the curvature lines are given by the straight line generators and their orthogonal trajectories.