Differential Geometry 13 14
Li,Zhi
6/10/2022
Isometric mapping
Isometric mapping is length-preserving. A mapping is isometric iff it is conformal and area-preserving.
Two surfaces are said to be applicable to one another if one can be deformed into the other by bending, with no stretching, compressing or tearing.
Envelope
For example, \((x-a)^2+y^2-1=0\) is a family of circles. Write into general form \((x-a)^2+(y-0)^2=1\), we see they are centered at point \(\{a,0\}\) with radius \(r=1\). Lines \(y=\pm 1\) are tangent to every circle of the family.
# initialize a plot
plot(c(-1, 5), c(-1.5, 1.5), type = "n")
# prepare "circle data"
radius = 1
center_x = 0#a=0
center_y = 0
theta = seq(0, 2 * pi, length = 200) # angles for drawing points around the circle
# draw the circle
lines(x = radius * cos(theta) + center_x, y = radius * sin(theta) + center_y)
center_x = 1#a=1
center_y = 0
theta = seq(0, 2 * pi, length = 200) # angles for drawing points around the circle
# draw the circle
lines(x = radius * cos(theta) + center_x, y = radius * sin(theta) + center_y)
x=1:5
envelope1 <- c(1,1,1,1,1)
envelope2 <- c(-1,-1,-1,-1,-1)
abline(h=1)
abline(h=-1)
Envelope of A Family of Surfaces
An Envelope is a surface that tangent to all surfaces in a family.
Say we have a short length helix on a cylinder surface, we make planes tangent to the cylinder surface along the helix. The tangent planes is a family of surfaces/planes. The cylinder surface along the helix is the envelope of the family of planes.
Characteristic of A Family of Surfaces
The characteristic is a line as 2 tangent planes get very close in our cylinder example above. But when 2 tangent planes really coincide each other, we get a plane not a line or curve. So the characteristic line or curve is an intersection of 2 planes or surfaces as they get very close to each other without coincidence, it is a limiting process.
https://encyclopedia2.thefreedictionary.com/Family+of+Surfaces
https://www.liquisearch.com/envelope_mathematics/envelope_of_a_family_of_surfaces