Michaelmas Flashcards
curvature k at a point of the curve is independent of the
parametrisation
Four vertex theorem
always at least four vertices on a smooth regular simple closed plane curve
principal curvatures
maximum and minimum if curvatures at a point of a surface
product of 2 principal curvatures at a point is the
Gaussian curvature
a is a smooth curve if all component functions are
smooth maps
the image of the interval under a curve is called
the trace of the curve
regular parametrised curve
derviative is not 0
unit speed curve
vector length is 1 everywhere. also called arc length parametrised curve
a parameter change for a smooth regular curve is
a map such that h is smooth, derivative of h is not 0, h(J) = I
reparametrisation is orientation preserving if
h’ > 0 and orient reversing if h’ < 0
length of smooth regular curve =
length a reparametrisation of curve
unit normal vector of a smooth regular plane curve is obtained by
anti-clockwise rotation of the unit tangent vector of the curve at pi / 2
the vector t’(s) of a smooth unit speed plane curve is parallel to
vector n(s)
signed curvature of a unit speed plane curve is
t’(s) = k(s)*n(s)
if a curve turns left curvature is
positive. if turns right its curvature is -ve.
point, alpha(u(0)), is an inflection point of smooth regular plane curve with curvature k if ………. and vertex if ….
k(u(0)) = 0 ,k’(u(0)) = 0
isoperimetric inequality: (length of smooth regular simple closed plane curve)^2 >=
4piarea of domain enclosed by curve. equal only if curve describes a circle
Fundamental theorem of local theory of plane curves
given an open interval and a smooth function, s(0), there exists a unique smooth unit speed plane curve, alpha, with curvature k where alpha(s(0)) = a and alpha’(s(0)) = v(0)
evolute is singular iff
curve is a vertex
if evolute is regular then its tangent vector is parallel to
the unit normal vector
if b is involute of a then
ap is evolute of curve b
vector product is
orthogonal to each vector, antisymmetric and will form a +vely oriented orthonormal basis if vectors are orthogonal
if curvature is not 0 the derivaitve of the binormal vector of curve is parallel to
the principal normal vector
curvature measures the rate the curve is
bending away from its straights tangent line
the plane through the curve spanned by its tangent and normal vectors is the
osculating plane of the curve
binormal vector is a unit normal the the osculating plane of a curve so derivative of binormal vector measures
the rate of change of the osculating
torsion measures the rate at which the curve is
twisting away from its osculating plane
if a smooth regular space curve with nowhere vanishing curvature and an affine plane exists a plane through the origin then
torsion - 0
Fundamental theorem of local theory of space curves: for any pair of smooth functions there exists a smooth unit speed space curve where the curve is unique up to ………….. so if another curve with same curvature and torsion that curve =
orientation preserving rigid motions.(x*rotation matrix + translation row vector) on curve
a subset is a regular surface for every point there exists an open set and a map if
the map is smooth and a homeomorphism aand partial derivatives are linearly independent
linearly independent can not be
a linear combination of each other
homeomorphism
bijective, continuos and has a continuous inverse
graph of a function of a surface, graphs(smooth function)
graph = (u, v, smooth function(u, v)) and is a regular surface
U is an open set and f a smooth function andc a regular value. then the preimage is a
regular surface
smooth map is diffeomorphisn if it is
bijective and inverse is smooth
surface of revolution is obtained by
rotating a curve in the x-z coordinate plane around the z-axis
curves obtained by rotating curve by a fixed angle are
meridans
tangent vector to a regular surface at a point on surface is a
tangent vector curve’(0) of a smooth curve
U is open, f is a smooth map and c a regular value of f. p is pre-image of c. the tangent plane T(p)S is the Euclidean plane which is
orthogonal to gradf(p)
canonical inner product has properties
bilinearity, symmetry and positivity
map is smooth at point of regular surface,S, if there exists a
local parametrisation, x:U -> S, with point in x(q) and q in U such that compostion f o x is smooth at q
Gauss map of the surface S is a smooth map which assigns to every point
a unit normal vector that is orthogonal to T(p)S
a surface is non-orietnable if there exists
no global Gauss map - no way to define the map globally continuously on surface.
if surface admits a global Gauss map the surface is
orientable
if f is a local isometry and a diffeomorphism then f is
an isometry and surfaces are isometric
conformal diffeomorphism
both a conformal map and a diffeomorphism
f a conformal map preserves
anlges between tangent vectors
Smooth function is
Differentiable everywhere
