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Elements of Map Projections
Published in Terry A. Slocum, Robert B. McMaster, Fritz C. Kessler, Hugh H. Howard, Thematic Cartography and Geovisualization, 2022
Terry A. Slocum, Robert B. McMaster, Fritz C. Kessler, Hugh H. Howard
A developable surface is a simple mathematical surface that can be flattened to form a plane without compressing or stretching any part of the original surface. There are three developable surfaces: cylinders, cones, and planes. To project the graticule from the reference globe to a map, first, a developable surface is conceptually placed over the reference globe, touching it along either a meridian or parallel or at one point (Figure 8.2); second, the graticule and geographic landmasses are projected onto the developable surface. Afterward, the developable surface is “unrolled,” revealing the graticule and landmasses.
Map Projections
Published in Julio Sanchez, Maria P. Canton, William Perrizo, Space Image Processing, 2018
Julio Sanchez, Maria P. Canton
In the study of map projections, the notion of a developable surface relates to one that can be transformed into a plane without distortion. A sphere is not developable; to transform its surface into a plane, it must be pressed out of shape. Therefore, the cartographer first transfers location data from the sphere into a developable surface and then develops the surface into a plane. The developable surfaces are the cylinder, the cone, and the plane itself. Figure 7.1 depicts projections onto the three developable surfaces.
Five-Point Perspective
Published in Craig Attebery, The Complete Guide To Perspective Drawing, 2018
Now that the theory of how lines appear when projected on a hemisphere is understood, how are they drawn on a flat surface? The paper has only two dimensions; the hemisphere has three. So, before these lines can be plotted on the paper, the hemisphere must first be flattened. Flattening a round surface is a problem that has plagued cartographers for thousands of years. It is a problem that still exists today; it is a problem that cannot be solved. Flattening a spherical surface, without distortion, is impossible. If a ball was cut in half, and then flattened, the only way to accomplish this task would be to tear, stretch, or fold the ball. No matter the approach, the shape of the ball will suffer. A cone or cylinder can be flattened without tearing, folding, or stretching. This is called a developable surface. A sphere is not a developable surface. Any flat representation of a sphere will, by its nature, be distorted.
Tangential developable and hydrodynamic surfaces for early stage of ship shape design
Published in Ships and Offshore Structures, 2023
A surface is called a developable surface if it can be developed on a plane without any lap fold or break. During this parabolic bending, the length of the curves and the angles between two curves belonging to the developable surface remain unchanged. Any developable surface is a cylindrical surface, or a conical surface, or else a surface of tangent lines of arbitrary space curve (torse). They are ruled surfaces with only parabolic points in which K = k1k2 = 0. So, an equality of Gaussian curvature K to zero is the sufficient and necessary condition for a developable surface.
Improved bolus shaping accuracy using the surface segmentation and spectral clustering
Published in International Journal of Modelling and Simulation, 2021
Rui Li, Qingjin Peng, Harry Ingleby, David Sasaki
After segmenting a 3D surface into several pieces based on the clustering results and smoothed boundary, 3D pieces are unfolded into 2D patches for bolus shaping. Only developable surfaces such as planes, generalized cylinders, and conical surfaces can be unfolded without deformation and distortion measured in the distance, angle and area. For non-developable surfaces, the unfolded surface will have some deformation and distortion. It is to search for a solution with the least deformation and distortion.