Math and Art final
Prespective Rule: 1
the farther an object is from the drawing plane, the smaller its perspective image in the drawing
Prespective Rules: 2
Lines that are mutually parallel and parallel to the drawing plane are depicted as parallel.
Prespective Rule: 3
mutually parallel lines that are not parallel to the drawing plane are depicted as intersecting line. The intersection point is called the vanishing point
Prespective rule: 4
Horizon line: if three or more classes of parallel lines are all parallel to a fixed plane that is not parallel to the drawing plane, and if these classes of parallel lines determine three or more vanishing points then all of these vanishing points occur on a single line. Moreover, if all lines are horizontal, the line is called the horizon line.
Regular convex polyhedron
a convex polyhedron is regular if all of the bounding polygona are congruent regular polygons and if each vertex is adjacent to the same number of bounding polygons
platonic solid:
a regular convex polyhedron
5 platonic solids
tetrahedron, cube, octahedron, dodecahron, isosahedron
Euler characteristic
if F is the number of faces of a polyhedron, E is the number of Edges and V is the number of vertices, then the valie of F-E+V is the Euler charactristuc of the polyhedron.
Euler charcteristic formula
F-E+V
are semi regular polyhedra platonic solids
no they are non-platonic solids
Semi regualr Polyhedra
A convex polyhedron is semiregular if all the bounding polygons are regular polygons(possibly more than one type) with edges the samd length and if each vertex is adjacent to the same number of bounding polygons, and there exists a fixed cyclic order of the types of polygons around yhe vertices
Archimedean solids
there are 13 Archimedean solid which are semiregular polyhedra. All prisims and antiprisms are semiregular polyhedra
all convex polyhedra have the same Euler Characteristic, what is it?
it is 2
there are 3 parts of polyhedra what are they?
vertex, edge, and face
topology: Homotopy
two spaces are homotopic if we can continuously deform one of them into the other without cutting or pasting. This deformation is called homotopy.
Topology: Homotopy what about holes
cant add holes or destroy them
are A and P homotopic?
yes! they have the same number of genus
are H and B homotopic?
no, B has two genus and H has zero
two-manifolds:
A two manifold is a space that locally feel like the surface of the plane
orientable two-manifolds
theres clearly an inside part and outside part ex. a sphere
is a torus an orientable two-manifold?
Yes
Genus:
the genus of a two-manifold is the number of consecutive closed circular cuts we can make on the suface without disconnecting it.
tip: however many holes you have thats how many cuts you can make
is the mobius strip a two-manifold?
no, it is a non-orientable surface
what is the formula for the euler characteristic of a two manifold?
V-E+F, not to be confused with platonic solids formula which is F+V-E
V = number of vertices
E = number of edges
F = number of polygon faces in ANY tiling of the surface
Euler Characteristic: shapes with holes formula
If x is a surface, denote the Euler characteristic of X by e(X) and denote the genus of X by g (X) Then
e(X) = 2-2g(x)
classification of two manifolds
every orientable two-manifold is homotopic to a sphere, torus, or a connected sum of (any finite number of) tori.
every non-orientable two manifold is homotopic to a projective plane or to a connected sum of (any finite number of) projective planes.
Euler Characteristic:formulas
1) F-E+V - platonic solids or plationic solid
2) V-E+F - two manifold
3) e(X) = 2-2g (X) note. X is the space (shape) and g is the genus). e is the EC ( euler characteristic
Ellipse definition:
if we cut the cone with a plane that intersects all the slant heights, the resulting shape is an ellipse.
note: ellipses are the circles verson of a rectangle
Circle definition:
if we cut the cone with a plane that intersects all the slant heights and is perpendicular to the axis, the resulting shape is an circle. A circle is a special case of an ellipse.
parabola definition:
if we cut the cone with a plane that is parallel to a tangent plane, the resulting shape is an parabola.
Hyperbola definiton:
if we cut the double cone with a plane that intersects both nappe, the resulting shape is an hyperbola.
Recall Euclid's Fifth Postulate:
For every line l and point P that does not like on l there exists a unqiue line m through P and parallel to l.
circle inversion rule:
the closer the point to center the farther out