Articles of geometry

Is this Cornu spiral positively oriented or not?.

Cornu Spiral I am learning about plane curves and I am told that if the curvature $\kappa$ is a linear function of arc length $s$. i.e $\kappa = s$ we obtain the Cornu Spiral. I find this difficult to understand for the following reason: If we take the $x$ axis in the above diagram to […]

How do you determine if a point sits inside a polygon?

Given the coordinates of a point $(x, y)$, what is a procedure for determining if it lies within a polygon whose vertices are $(x_1, y_1), (x_2, y_2), \ldots , (x_n,y_n)$?

Integer coordinate set of points that is a member of sphere surface

I have a graphic application to develop which involve many spheres. I should determine then on run time. Supposing that I have a sphere of radius r, how can I determine the sub set of the sphere surface points that are integer? E.g., $r = 10$ I can have $(10,0,0), (8,6,0),$ etc. (Obs.: I really […]

Do four dimensional vectors have a cross product property?

This question already has an answer here: Cross product in $\mathbb R^n$ 2 answers

Simple prove that product of the diagonals of a polygon = N

There is a beautiful fact: If you take a regular N-sided polygon, which is inscribed in the unit circle and find the product of all its diagonals (including two sides) carried out from one corner you will get N exactly: $A_1A_2\cdot A_1A_3\cdot …\cdot A_1A_N = N$ For example, for a square we have $\sqrt{2}\cdot 2\cdot […]

Geometric meaning of the determinant of a matrix

What is the geometric meaning of the determinant of a matrix? I know that “The determinant of a matrix represents the area of ​​a rectangle.” Perhaps this phrase is imprecise, but I would like to know something more, please. Thank you very much.

V.I. Arnold says Russian students can't solve this problem, but American students can — why?

In a book of word problems by V.I Arnold, the following appears: The hypotenuse of a right-angled triangle (in a standard American examination) is 10 inches, the altitude dropped onto it is 6 inches. Find the area of the triangle. American school students had been coping successfully with this problem for over a decade. But […]

Inscribing a rhombus within a convex quadrilateral

I was wondering if it is possible to inscribe a rhombus within any arbitrary convex quadrilateral using only compass and ruler? If possible, could you describe the method? If not could you give an example of a convex quadrilateral which one can not inscribe a rhombus within?

Inequality involving sides of a triangle

How can one show that for triangles of sides $a,b,c$ that $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} < 2$$ My proof is long winded, which is why I am posting the problem here. Step 1: let $a=x+y$, $b=y+z$, $c=x+z$, and let $x+y+z=1$ to get $\frac{1-x}{1+x}+\frac{1-y}{1+y}+\frac{1-z}{1+z}<2$ Step 2: consider the function $f(x)=\frac{1-x}{1+x}$, and note that it is convex on the interval […]

What is a point?

In geometry, what is a point? I have seen Euclid’s definition and definitions in some text books. Nowhere have I found a complete notion. And then I made a definition out from everything that I know regarding maths. Now, I need to know what I know is correct or not. One book said, if we […]