1
vote
2answers
59 views

Total/Gaussian curvature is intrinsic, yet mean curvature is extrinsic, why?

What characteristics define the total/mean curvature to be intrinsic/extrinsic accordingly? What is different geometrically about these curvatures that cause them to be defined as this?
5
votes
1answer
48 views

Is there a rigorous definition of the term “coordinate system”?

You hear the term coordinate system thrown around a lot, and we all know the usual examples (polar coordinates in $\mathbb{R}^2$, spherical coordinates in $\mathbb{R}^3$, etc.), but in truth I have no ...
1
vote
2answers
37 views

Point as an element of an affine space vs point as an element of a topological space?

I am searching for the "most natural" definition of a (geometrical/space) point as an element of "something" in mathematics (I am trying to design a small computational geometry library on strong ...
1
vote
1answer
52 views

Definition of “point” and “vector” in $\Bbb{R}^n$, and a model for $\Bbb{A}^n(\Bbb{R})$..

Can I use the following definitions? let $(a_1,a_2,..,a_{n+1}) \in \Bbb{R}^{n+1}$, $(a_1,a_2,..,a_{n+1})$ is a point of $\Bbb{R}^n$ if $$a_1=1 \wedge \forall i \in \{2,...,n+1\} (a_i \in \Bbb{R})$$ ...
0
votes
0answers
20 views

Adjacency corner case

If two rectangles touch, but only touch at one of their corners (example, Rectangle A's upper-right corner is touch Rectangle B's lower-left corner), are they adjacent to one another? Why or why not?! ...
0
votes
0answers
17 views

Proper vs Subline Adjacency

Can someone explain to me the technical difference between "proper" and "subline" adjacency? The only definition I have is as follows: Adjacency may be proper or subline, where a subline adjacency ...
8
votes
1answer
253 views

What is the exact and precise definition of an ANGLE?

On wikipedea I found a definition of an Angle as such: "In order to measure an angle θ, a circular arc centered at the vertex of the angle is drawn, e.g. with a pair of compasses. The length of ...
2
votes
0answers
28 views

What are defintions of ellipticity?

Ellipticity seems to have many definitions. So far, I'm aware of three. Are there other definitions of ellipticity that you know of, and if so, where did you encounter them? The three definitions ...
1
vote
2answers
48 views

definition of rectangle

I am wondering whether it is fine to define a rectangle with right angles like Wikipedia's page of rectangle. In Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. ...
4
votes
4answers
229 views

Confusion about the usage of points vs. vectors

As far as definitions go, understand the difference between a vector and a point. A vector can be translated and still be the same vector, whereas a point is fixed. But I would like some clarification ...
8
votes
3answers
310 views

What does area represent?

Since any two Euclidean shapes have an infinite number of points inside of them, shapes with different area have the same infinite number of points in them (and any object has the same number of ...
1
vote
3answers
178 views

Cantor Set and Fractals

I have read that the Cantor set is considered a fractal. I am referring to the Cantor set in which the middle third of a real line is removed recursively. I see that this is recursively defined, but ...
7
votes
3answers
227 views

What's Geometry?

I am a grad student. I am writing an article on geometry and relativity theory and trying to start with discussing basic ideas of topology. In my article I tried very hard to motivate the idea of ...
9
votes
1answer
210 views

Is there a way to relate convexity to Gaussian curvature?

This is a vague question because I'm not sure what I want to ask. An ellipsoid has positive curvature everywhere, and bounds a convex subset of $\mathbb R^3$. What I want to say now is "It seems as ...
-4
votes
2answers
750 views

Geometric Definitions: What is a straight line? What is a circle?

What is a straight line? I need a geometric definition of it. The equation of a straight line is known to me.I am saying about a straight line of 2D plane. What is a circle? I need a geometric ...
3
votes
3answers
75 views

Definition of a tangent

I've been involved in a discussion on definition of a tangent and would appreciate a bit of help. At my high school and at my college I was taught that a definition of a tangent is 'a line that ...
7
votes
4answers
413 views

What is straight line?

I have found the definition of line in metric space. It is general but has two problems. Considering about $\mathbb R^2$ equipped rectilinear distance, every line by this definition contains a ...
3
votes
0answers
78 views

Closed / embedded surface

Given a closed surface in $\mathbb R^3$, is it necessarily an "embedded surface"? I think it is true, but that is just because I can't think of a closed surface for which we cannot construct a smooth ...
5
votes
2answers
316 views

the definition of the area of a surface

When we say the area of a rectangle is the product of the length by the width is it a definition based on geometric intuition or is it a result? I know it is a result that we can find after defining ...
1
vote
3answers
70 views

A parametrized surface

If I am given a map $f:U\subset \mathbb R^2 \to \mathbb R^3$ where $(x,y)\to (f(x,y),g(x,y),h(x,y))$. Is this necessarily a "parametrized surface"?  Am I right in thinking that any map of the above ...
2
votes
1answer
39 views

Terminology clarification: ***exchanges***

I need help with a terminology definition. If we say "R is a reflection that exchanges the sides a and b in some triangle", does it mean sides a and b have the same length and the reflection maps one ...
0
votes
1answer
120 views

Motivation behind the definition of reflections in affine hyperplane

What is the motivation behind the definition of the reflection map in affine hyperplanes? $R: x \to x-2(x\cdot u-c)u$ where $u\cdot x=c$ defines the affine plane. Of course one requirement is for it ...
1
vote
4answers
270 views

What is the name for a maximal convex set of points contained in another set of points?

What is the name for a maximal convex set of points contained in another set of points X? Maximal in terms of inclusion. For the desired set to be unique, X can be restricted to be a simple polygon ...
2
votes
3answers
147 views

How can the geometry (and the reals) be motivated from the bottom up?

I'm really not sure that I know what I'm talking about, or if I should just go and learn more math before questioning such things, but I'd like to have answers to the following questions that don't ...
1
vote
3answers
181 views

After saw this piece of discussion, i ask myself what is the most rigorous definition of the circle, but i can't figure it out

Discussion: Is value of $\pi$ = 4? so what is the "real definition" of a circle? i think the original solution from wikipedia is too ambigous, i couldn't find why the circumference of the circle is ...
5
votes
2answers
758 views

Embedding, immersion

Could someone please explain what "embedding" means? (Maybe a more intuitive definition) I read that the Klein bottle and real projective plane cannot be embedded in ${\mathbb R}^3$ but is embedded in ...
19
votes
3answers
2k views

Is a line parallel with itself?

Simple Question, but I'm finding a lot of dispute on the "lesser" internet. Basically, given a line, is it parallel with itself?
6
votes
3answers
1k views

What is the proper geometric description of a the oval used for a horse racetrack?

I'm talking about the shape made up of a rectangle with a semi-circle at each end. Does it have a particular name?