Shortest algorithm to determine sub-triangle

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
0
down vote

favorite












Fun (?) question here. Given an arbitrary point on the triangle pictured below, what is the "best" way to determine which of the 60 sub triangles it lies within? Suppose I have the coordinates for all sub-triangle vertices.



"Best" is from a programming POV, so my though is to just check if the point is above, below (or on), a given line... then repeat. So maybe you check if it's above or below the 2nd steepest of the 4 lines that intersect at the bottom left vertex - if it's above, check above/below line x, if it's below check above/below line y, etc.



With minimal thought put into the algorithm, I was able to determine the location in 7 steps (meaning above/below 7 different lines) for 16 of the sub-triangles, 6 steps for 32 of the sub-triangles, and 5 steps for 12 of the sub-triangles. I'm fairly certain that this is not optimal. When I say optimal, I'd really like to minimize the max, so the 7 step triangles bug me.



I think 6 steps for 56 of the triangles and 5 steps for 4 of the triangles is a lower bound on the length of the "optimal" solution, but I don't think is achievable.



Any thoughts at the optimal solution? My gut said to split the sub triangles as closely to evenly as possible in each step, but I'm not even sure that's true (or how to do that).



In case it helps, you can view the big triangle as 15 medium sized triangles (3 in each of the 3 corners, 6 in the middle). Each of those 15 medium sized triangles is subdivided into 4 small triangles.



Thanks!



The Triangle




geometry algorithms




share|cite|improve this question















  • 1




    Try posting this on Code Golf SE. Mathematics SE doesn't really provide 'algorithms' which you ask for.
    – Shawn Li
    Aug 1 at 20:53











  • I will, and thanks for the advice. However, I think the crux of the problem is mathematical in nature as I'm looking to exploit the symmetries of this triangle.
    – Brian
    Aug 1 at 21:05






  • 2




    It seems move like stack overflow or computerscience SE than code golf to me. Code golf is about writing programs in the fewest possible number of bits.
    – saulspatz
    Aug 1 at 21:28











  • My guess is that the answer depends on how the triangles are constructed. Do you have that information? Or is it just some random partition into 60 triangles?
    – Andrei
    Aug 1 at 23:04






  • 2




    @ShawnLi Please do not suggest users to go to any site you're unfamiliar of. We only allow programming challenge on our site, with some exceptions. Question about constructing an algorithm is not one of them.
    – user202729
    Aug 2 at 0:52














up vote
0
down vote

favorite












Fun (?) question here. Given an arbitrary point on the triangle pictured below, what is the "best" way to determine which of the 60 sub triangles it lies within? Suppose I have the coordinates for all sub-triangle vertices.



"Best" is from a programming POV, so my though is to just check if the point is above, below (or on), a given line... then repeat. So maybe you check if it's above or below the 2nd steepest of the 4 lines that intersect at the bottom left vertex - if it's above, check above/below line x, if it's below check above/below line y, etc.



With minimal thought put into the algorithm, I was able to determine the location in 7 steps (meaning above/below 7 different lines) for 16 of the sub-triangles, 6 steps for 32 of the sub-triangles, and 5 steps for 12 of the sub-triangles. I'm fairly certain that this is not optimal. When I say optimal, I'd really like to minimize the max, so the 7 step triangles bug me.



I think 6 steps for 56 of the triangles and 5 steps for 4 of the triangles is a lower bound on the length of the "optimal" solution, but I don't think is achievable.



Any thoughts at the optimal solution? My gut said to split the sub triangles as closely to evenly as possible in each step, but I'm not even sure that's true (or how to do that).



In case it helps, you can view the big triangle as 15 medium sized triangles (3 in each of the 3 corners, 6 in the middle). Each of those 15 medium sized triangles is subdivided into 4 small triangles.



Thanks!



The Triangle




geometry algorithms




share|cite|improve this question















  • 1




    Try posting this on Code Golf SE. Mathematics SE doesn't really provide 'algorithms' which you ask for.
    – Shawn Li
    Aug 1 at 20:53











  • I will, and thanks for the advice. However, I think the crux of the problem is mathematical in nature as I'm looking to exploit the symmetries of this triangle.
    – Brian
    Aug 1 at 21:05






  • 2




    It seems move like stack overflow or computerscience SE than code golf to me. Code golf is about writing programs in the fewest possible number of bits.
    – saulspatz
    Aug 1 at 21:28











  • My guess is that the answer depends on how the triangles are constructed. Do you have that information? Or is it just some random partition into 60 triangles?
    – Andrei
    Aug 1 at 23:04






  • 2




    @ShawnLi Please do not suggest users to go to any site you're unfamiliar of. We only allow programming challenge on our site, with some exceptions. Question about constructing an algorithm is not one of them.
    – user202729
    Aug 2 at 0:52












up vote
0
down vote

favorite









up vote
0
down vote

favorite











Fun (?) question here. Given an arbitrary point on the triangle pictured below, what is the "best" way to determine which of the 60 sub triangles it lies within? Suppose I have the coordinates for all sub-triangle vertices.



"Best" is from a programming POV, so my though is to just check if the point is above, below (or on), a given line... then repeat. So maybe you check if it's above or below the 2nd steepest of the 4 lines that intersect at the bottom left vertex - if it's above, check above/below line x, if it's below check above/below line y, etc.



With minimal thought put into the algorithm, I was able to determine the location in 7 steps (meaning above/below 7 different lines) for 16 of the sub-triangles, 6 steps for 32 of the sub-triangles, and 5 steps for 12 of the sub-triangles. I'm fairly certain that this is not optimal. When I say optimal, I'd really like to minimize the max, so the 7 step triangles bug me.



I think 6 steps for 56 of the triangles and 5 steps for 4 of the triangles is a lower bound on the length of the "optimal" solution, but I don't think is achievable.



Any thoughts at the optimal solution? My gut said to split the sub triangles as closely to evenly as possible in each step, but I'm not even sure that's true (or how to do that).



In case it helps, you can view the big triangle as 15 medium sized triangles (3 in each of the 3 corners, 6 in the middle). Each of those 15 medium sized triangles is subdivided into 4 small triangles.



Thanks!



The Triangle




geometry algorithms




share|cite|improve this question











Fun (?) question here. Given an arbitrary point on the triangle pictured below, what is the "best" way to determine which of the 60 sub triangles it lies within? Suppose I have the coordinates for all sub-triangle vertices.



"Best" is from a programming POV, so my though is to just check if the point is above, below (or on), a given line... then repeat. So maybe you check if it's above or below the 2nd steepest of the 4 lines that intersect at the bottom left vertex - if it's above, check above/below line x, if it's below check above/below line y, etc.



With minimal thought put into the algorithm, I was able to determine the location in 7 steps (meaning above/below 7 different lines) for 16 of the sub-triangles, 6 steps for 32 of the sub-triangles, and 5 steps for 12 of the sub-triangles. I'm fairly certain that this is not optimal. When I say optimal, I'd really like to minimize the max, so the 7 step triangles bug me.



I think 6 steps for 56 of the triangles and 5 steps for 4 of the triangles is a lower bound on the length of the "optimal" solution, but I don't think is achievable.



Any thoughts at the optimal solution? My gut said to split the sub triangles as closely to evenly as possible in each step, but I'm not even sure that's true (or how to do that).



In case it helps, you can view the big triangle as 15 medium sized triangles (3 in each of the 3 corners, 6 in the middle). Each of those 15 medium sized triangles is subdivided into 4 small triangles.



Thanks!



The Triangle





geometry algorithms





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Aug 1 at 20:51









Brian

287




287







  • 1




    Try posting this on Code Golf SE. Mathematics SE doesn't really provide 'algorithms' which you ask for.
    – Shawn Li
    Aug 1 at 20:53











  • I will, and thanks for the advice. However, I think the crux of the problem is mathematical in nature as I'm looking to exploit the symmetries of this triangle.
    – Brian
    Aug 1 at 21:05






  • 2




    It seems move like stack overflow or computerscience SE than code golf to me. Code golf is about writing programs in the fewest possible number of bits.
    – saulspatz
    Aug 1 at 21:28











  • My guess is that the answer depends on how the triangles are constructed. Do you have that information? Or is it just some random partition into 60 triangles?
    – Andrei
    Aug 1 at 23:04






  • 2




    @ShawnLi Please do not suggest users to go to any site you're unfamiliar of. We only allow programming challenge on our site, with some exceptions. Question about constructing an algorithm is not one of them.
    – user202729
    Aug 2 at 0:52












  • 1




    Try posting this on Code Golf SE. Mathematics SE doesn't really provide 'algorithms' which you ask for.
    – Shawn Li
    Aug 1 at 20:53











  • I will, and thanks for the advice. However, I think the crux of the problem is mathematical in nature as I'm looking to exploit the symmetries of this triangle.
    – Brian
    Aug 1 at 21:05






  • 2




    It seems move like stack overflow or computerscience SE than code golf to me. Code golf is about writing programs in the fewest possible number of bits.
    – saulspatz
    Aug 1 at 21:28











  • My guess is that the answer depends on how the triangles are constructed. Do you have that information? Or is it just some random partition into 60 triangles?
    – Andrei
    Aug 1 at 23:04






  • 2




    @ShawnLi Please do not suggest users to go to any site you're unfamiliar of. We only allow programming challenge on our site, with some exceptions. Question about constructing an algorithm is not one of them.
    – user202729
    Aug 2 at 0:52







1




1




Try posting this on Code Golf SE. Mathematics SE doesn't really provide 'algorithms' which you ask for.
– Shawn Li
Aug 1 at 20:53





Try posting this on Code Golf SE. Mathematics SE doesn't really provide 'algorithms' which you ask for.
– Shawn Li
Aug 1 at 20:53













I will, and thanks for the advice. However, I think the crux of the problem is mathematical in nature as I'm looking to exploit the symmetries of this triangle.
– Brian
Aug 1 at 21:05




I will, and thanks for the advice. However, I think the crux of the problem is mathematical in nature as I'm looking to exploit the symmetries of this triangle.
– Brian
Aug 1 at 21:05




2




2




It seems move like stack overflow or computerscience SE than code golf to me. Code golf is about writing programs in the fewest possible number of bits.
– saulspatz
Aug 1 at 21:28





It seems move like stack overflow or computerscience SE than code golf to me. Code golf is about writing programs in the fewest possible number of bits.
– saulspatz
Aug 1 at 21:28













My guess is that the answer depends on how the triangles are constructed. Do you have that information? Or is it just some random partition into 60 triangles?
– Andrei
Aug 1 at 23:04




My guess is that the answer depends on how the triangles are constructed. Do you have that information? Or is it just some random partition into 60 triangles?
– Andrei
Aug 1 at 23:04




2




2




@ShawnLi Please do not suggest users to go to any site you're unfamiliar of. We only allow programming challenge on our site, with some exceptions. Question about constructing an algorithm is not one of them.
– user202729
Aug 2 at 0:52




@ShawnLi Please do not suggest users to go to any site you're unfamiliar of. We only allow programming challenge on our site, with some exceptions. Question about constructing an algorithm is not one of them.
– user202729
Aug 2 at 0:52















active

oldest

votes











Your Answer




StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");

StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);

else
createEditor();

);

function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
convertImagesToLinks: true,
noModals: false,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);



);








 

draft saved


draft discarded


















StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2869507%2fshortest-algorithm-to-determine-sub-triangle%23new-answer', 'question_page');

);

Post as a guest






















active

oldest

votes













active

oldest

votes









active

oldest

votes






active

oldest

votes










 

draft saved


draft discarded


























 


draft saved


draft discarded














StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2869507%2fshortest-algorithm-to-determine-sub-triangle%23new-answer', 'question_page');

);

Post as a guest
































































Comments

Post a Comment

Popular posts from this blog

What is the equation of a 3D cone with generalised tilt?

Image
Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite What is the equation of a 3D cone with generalized tilt? I've noticed that in most equations given to represent a cone, there is no parameter which defines the tilt of the cone in 3D space and that most of them have their point at the origin $(0,0,0)$ - I was wondering if anyone could give me a more generalized cone equation for a cone in any position in 3D space 3d share | cite | improve this question edited Jul 25 '16 at 0:05 ervx 9,425 3 13 36 asked Jul 24 '16 at 15:38 Charlene 11 2 2 Welcome to Math.SE! Could you please explain a few things to help people give an answer useful to you: 1. What do you mean by "generalized tilt"? 2. Are you asking about a right circular cone? Does the angle at the vertex matter? 3. What form of answer (implicit, parametric...?) are you looking for? 4. If this is homework, what tools do you have available, an...
Read more

Color the edges and diagonals of a regular polygon

Image
Clash Royale CLAN TAG #URR8PPP up vote 5 down vote favorite 1 Here is the problem: For what $n$ is it possible to color the edges and diagonals of an $n$-side regular polygon with $dfracbinomn23$ colors, such that you use every color exactly three times and for every color the three segments (edges or diagonals) with that color form a triangle? Trivially $nequiv 0 text or 1 pmod3$. I can also prove that if the statement is true for $k$ than it is true for $3k$ as well. How to finish? Please help! Thanks combinatorial-geometry share | cite | improve this question edited Aug 6 at 21:57 alcana 118 4 asked Aug 6 at 21:15 Leo Gardner 356 11 Even assuming "polyhpn" in the title is a typo for polygon , it is unclear what you mean by "the edges and sides". Aren't the side of a regular polygon the same as the edges? A regular $n$-sided polygon has $n$ edges. – hardmath Aug 6 at 21:20 3 $n$ ne...
Read more

Relationship between determinant of matrix and determinant of adjoint?

Image
Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite We are studying adjoints in class, and I was curious if there is a relationship between the determinant of matrix A, and the determinant of the adjoint of matrix A? I assume there would be a relationship because finding the adjoint requires creating a cofactor matrix and then transposing it. linear-algebra determinant adjoint-operators share | cite | improve this question asked Nov 17 '17 at 17:32 CluelessCoder 32 5 For a square matrix $A$ of order $n$, we have $A(operatornameadj A)=(operatornameadj A)A=|A|I_n$ where $I_n$ is the identity matrix of order $n$. This should tell you about the determinant of the adjoint in terms of that of $A$ (use multiplicative property and other elementary properties of determinants). – Prasun Biswas Nov 17 '17 at 17:35 1 @CluelessCoder: see here: math.stackexchange.com/questions/516127/…...
Read more