Developable surface

In mathematics, a developable surface (historically called a torse) is a smooth surface with zero Gaussian curvature. That is, it is a surface that can be flattened onto a plane without distortion (i.e. it can be bent without stretching or compression). Conversely, it is a surface which can be made by transforming a plane (i.e. "folding", "bending", "rolling", "cutting" and/or "gluing"). Because of these properties, developable surfaces are widely used in the design and fabrication of items to be made from sheet materials, ranging from textiles to sheet metal such as ductwork to shipbuilding. Relatedly, in classical differential geometry, a development is the rolling of one smooth surface over another in Euclidean space. A surface is developable if it has a development. All developable surfaces embedded in 3D-space are ruled surfaces (though hyperboloids are examples of ruled surfaces which are not developable). Because of this, many developable surfaces can be visualised as the surface formed by moving a straight line in space. For example, a cone is formed by keeping one end-point of a line fixed whilst moving the other end-point in a circle. However, there are developable surfaces in four-dimensional space which are not ruled.

In mathematics, a developable surface (historically called a torse) is a smooth surface with zero Gaussian curvature. That is, it is a surface that can be flattened onto a plane without distortion (i.e. it can be bent without stretching or compression). Conversely, it is a surface which can be made by transforming a plane (i.e. "folding", "bending", "rolling", "cutting" and/or "gluing"). Because of these properties, developable surfaces are widely used in the design and fabrication of items to be made from sheet materials, ranging from textiles to sheet metal such as ductwork to shipbuilding. Relatedly, in classical differential geometry, a development is the rolling of one smooth surface over another in Euclidean space. A surface is developable if it has a development. All developable surfaces embedded in 3D-space are ruled surfaces (though hyperboloids are examples of ruled surfaces which are not developable). Because of this, many developable surfaces can be visualised as the surface formed by moving a straight line in space. For example, a cone is formed by keeping one end-point of a line fixed whilst moving the other end-point in a circle. However, there are developable surfaces in four-dimensional space which are not ruled. ==Properties== A surface defined by F(x, y, z) = 0 is developable if \left|\begin{array}{llll} F_{x x} & F_{x y} & F_{x z} & F_x \\ F_{y x} & F_{y y} & F_{y z} & F_y \\ F_{z x} & F_{z y} & F_{z z} & F_z \\ F_x & F_y & F_z & 0 \end{array}\right|=0, \quad F_x=\partial F / \partial x, \text { etc } Consider a ruled surface F(s, v)=c(s)+v d(s), \quad|d(s)|=1 with s the ruling parameter and 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 F(s, v) is the same for all v with that fixed s . In formulas, it says \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 c(s)+v_0(s) d(s) , where 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 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. (white) z=x^2-y^2,\; 0\le x,y\le 0.5 ]] 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. ==See also== Development (differential geometry) Developable roller ==References== ==External links== Examples of developable surfaces on the Rhino3DE website

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