Difference between revisions of "Formulas relating the number of lines, sections, and intersection points"
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| − | + | \text{Some formulas relating the number of intersections, lines, and sections in a plane.} | |
| − | + | \text{1) The maximum number of intersection points of n lines is <math>\frac{n(n-1)}{2}</math>.} | |
Proof: Each line intersects all the other lines to create the maximum number of intersections. The first line intersects n-1 lines, the second one has n-2 | Proof: Each line intersects all the other lines to create the maximum number of intersections. The first line intersects n-1 lines, the second one has n-2 | ||
new intersection points, and so on. | new intersection points, and so on. | ||
| − | + | \text{2) The maximum number of planar sections (assuming the plane is bound at some point) created by n lines is \frac{n^2+n+2}{2}} | |
Proof: Before there are any lines, there is one section. The first line passes through one section, splitting it in two. The second line passes through | Proof: Before there are any lines, there is one section. The first line passes through one section, splitting it in two. The second line passes through | ||
two sections, splitting it into a total of four sections. The third line passes through a maximum of three existing sections, adding three new sections to the | two sections, splitting it into a total of four sections. The third line passes through a maximum of three existing sections, adding three new sections to the | ||
plane. (someone turn this into a formal proof with induction please) | plane. (someone turn this into a formal proof with induction please) | ||
| − | + | \text{3)The number of lines + number of intersection points = number of sections -1 assuming that no three lines intersect at one point.} | |
The proof can be made by bashing the equation with the previous formulas | The proof can be made by bashing the equation with the previous formulas | ||
| − | + | \text{4) The number of curved lines (enclosed circles)+ number of lines+ number of intersection points -1= number of sections created, provided that no three curved or straight lines intersect at the same point.} | |
Proof: Almost the same, note however that two curved lines and intersect at two points, same for one curved and one straight line. | Proof: Almost the same, note however that two curved lines and intersect at two points, same for one curved and one straight line. | ||
Revision as of 17:17, 4 June 2013
\text{Some formulas relating the number of intersections, lines, and sections in a plane.}
\text{1) The maximum number of intersection points of n lines is 
.}
Proof: Each line intersects all the other lines to create the maximum number of intersections. The first line intersects n-1 lines, the second one has n-2
new intersection points, and so on.
\text{2) The maximum number of planar sections (assuming the plane is bound at some point) created by n lines is \frac{n^2+n+2}{2}}
Proof: Before there are any lines, there is one section. The first line passes through one section, splitting it in two. The second line passes through
two sections, splitting it into a total of four sections. The third line passes through a maximum of three existing sections, adding three new sections to the
plane. (someone turn this into a formal proof with induction please)
\text{3)The number of lines + number of intersection points = number of sections -1 assuming that no three lines intersect at one point.}
The proof can be made by bashing the equation with the previous formulas
\text{4) The number of curved lines (enclosed circles)+ number of lines+ number of intersection points -1= number of sections created, provided that no three curved or straight lines intersect at the same point.}
Proof: Almost the same, note however that two curved lines and intersect at two points, same for one curved and one straight line.
