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of all of its tangent planes. his is the basis for the representation shown in Fig.
.
(let). Every point on the surface is mapped into the two points representing its tan-
gent plane at the point. his generates two planar regions and N
suchregions
in N-D. hese regions are linked, just like the polygons above, to provide the N
−
points representing eachtangent hyperplaneandtherefore reconstruct the hypersur-
face. Classes of surfaces can be immediately distinguished by their
−
-coords displays
(seeHungandInselberg (
)andInselberg(
)formorerecent results).Forde-
velopable surfaces, the regions consist of boundary curves only (no interior points);
the regions for ruled surfaces have grids consisting of straight lines; while quadric
surfaces have regions with conic boundaries. hese are just some examples.
hereisasimplerbutinexactsurfacerepresentation thatisquiteusefulwhenused
judiciously. he polygonal lines representing points on the boundary are plotted,
and their envelope “represents” the surface; the “ ” are a reminder that this is not
a unique representation. Figure
.
(let) shows the upper and lower envelopes for
a sphere in
-D, which consist of four overlapping hyperbolae; this must be distin-
guished from that in Fig.
.
(right), which is exact as determined by the sphere's
tangent planes. By retaining the exact surface description (i.e., its equation) inter-
nally,interiorpointscanbeconstructedanddisplayed,asshownforthe
-Dspherein
Fig.
.
(let). he same construction is shown on the right, but for a more complex
-dimensional convex hypersurface(“model”).heintermediate curves (upperand
lower) also provide valuable information and preview coming attractions. heyindi-
cate the neighborhood of the point (represented by the polygonal line) and provide
afeelforthelocal curvature.Notethe narrowstrips(ascomparedtothesurrounding
ones),whichindicate the critical variables wherethe point is bumping the boundary.
A theorem guarantees that a polygonal line which is in-between all of the intermedi-
ate curves/envelopes represents an interior point of the hypersurface,and all interior
points can be found in this way. If the polygonal line is tangent to any one of the in-
termediate curves then itrepresents aboundary point, whileit represents an exterior
point if it crosses any one of the intermediate curves. he latter enables us to see, in
Figure
.
.
(Let)Asmoothsurfaceσ is represented by two planar regions σ
, σ
′
consisting of
pairs of points that represent its tangent planes. (Right) One of the two hyperbolic regions representing
aspherein
-D
