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Math Help - Stars

  1. #1
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    Stars

    I'm trying to program a 5 pointed star and I'm having a hard time trying to understanding how to go about it. The program needs to be as such that I can just input a side legnth and it will create a star of that size.

    Are there any special realtionships concerning stars? I can understand that the vertices of the star will all be at angles of 36 around the centre but I can seem to see much else.

    Thanks.
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  2. #2
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    Quote Originally Posted by Alyosha View Post
    I'm trying to program a 5 pointed star and I'm having a hard time trying to understanding how to go about it. The program needs to be as such that I can just input a side legnth and it will create a star of that size.

    Are there any special realtionships concerning stars? I can understand that the vertices of the star will all be at angles of 36 around the centre but I can seem to see much else.

    Thanks.
    If one sidelength of one of the 5 triangular Points" of the star is your basis, then:

    Let s = the sidelength.
    So the perimeter of the star is 5(2s) = 10*s

    Radius, R, of the circle circumscribing the whole star is
    R = s*cos(18deg) +s*sin(18deg)tan(54deg)
    R = (1.37638)*s

    "Wing" length, or from one vertex to another vertex, l, is
    L = s +2*s*sin(18deg) +s
    L = (2.61803)*s

    Then, a shortcut, divide the circumscribing circle into 5 equal arcs. So one arc is 360/5 = 72 degrees long. Or the central angle of each arc is 72 degrees.
    Then draw chords to connect those 5 points on the circumference.

    You think you can program now the 5-pointed star?
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  3. #3
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    Not sure...

    can you break this bit down for me a bit?

    Radius, R, of the circle circumscribing the whole star is
    R = s*cos(18deg) +s*sin(18deg)tan(54deg)
    R = (1.37638)*s

    where did you get the size of the angles from?
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  4. #4
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    Quote Originally Posted by Alyosha View Post
    I'm trying to program a 5 pointed star and I'm having a hard time trying to understanding how to go about it. The program needs to be as such that I can just input a side legnth and it will create a star of that size.

    Are there any special realtionships concerning stars? I can understand that the vertices of the star will all be at angles of 36 around the centre but I can seem to see much else.

    Thanks.
    The only value you need is the radius of the circumcircle.

    1. Place the 5 points on the perimeter of this cicle. The central angle is 72.

    2. Now draw straight lines starting at A:

    ACEBDA

    3. See attachment
    Attached Thumbnails Attached Thumbnails Stars-pentagramm.png  
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  5. #5
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    Quote Originally Posted by Alyosha View Post
    Not sure...

    can you break this bit down for me a bit?

    Radius, R, of the circle circumscribing the whole star is
    R = s*cos(18deg) +s*sin(18deg)tan(54deg)
    R = (1.37638)*s

    where did you get the size of the angles from?
    Draw the figure, or sketch it.
    (I don't know how to draw in computers so I will just describe it)
    Draw it such that one of the 5 "star-points" is the topmost. Call that point, A.
    Draw a vertical line segment from A to the center of the star (or of the circumscribing circle). Call that center, point O.
    Draw a line segment from O to one of the bases (lower ends) of the isosceles triangle (the "star-point"). Call that end or "inside corner of the star" as point B.
    Draw a horizontal line segment connecting the two lower ends of the star-point. Call the intersection of that line segment with AO as point C.

    AC +CO = R ....the circumscribing radius.

    AB = s

    angle BAO = angle BAC = 36/2 = 18 degrees
    So, AC = s*cos(18deg) ---------**

    BC = s*sin(18deg)

    In right triangle BCO:
    angle OBC = 108/2 = 54 degrees -----that is because an inetrior angle of a regular pentagon is (5-2)(180deg)/5 = 108 deg. And angle OBC is half of one of the interior angles of the pentagon of the star.

    tan(54deg) = CO /BC
    CO = (BC)*tan(54deg)
    So,
    CO = [s*sin(18deg)]*tan(54deg)
    CO = s*sin(18deg)tan(54deg) ------**

    Therefore,
    R = s*cos(18deg) +s*sin(18deg)tan(54deg)

    Check your calculator if you can get R = (1.37638)*s
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  6. #6
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    Draw the figure, or sketch it.
    (I don't know how to draw in computers so I will just describe it)
    Draw it such that one of the 5 "star-points" is the topmost. Call that point, A.
    Draw a vertical line segment from A to the center of the star (or of the circumscribing circle). Call that center, point O.
    Draw a line segment from O to one of the bases (lower ends) of the isosceles triangle (the "star-point"). Call that end or "inside corner of the star" as point B.
    Draw a horizontal line segment connecting the two lower ends of the star-point. Call the intersection of that line segment with AO as point C.

    I'm not sure I understand this. AO runs from the topmost point of the star to the center, if draw a horizontal line segment connecting the two lower ends of the star-point, it doesn't intersect with AO as AO stops in the middle of the star.
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  7. #7
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    Quote Originally Posted by earboth View Post
    The only value you need is the radius of the circumcircle.

    1. Place the 5 points on the perimeter of this cicle. The central angle is 72.

    2. Now draw straight lines starting at A:

    ACEBDA

    3. See attachment
    The program needs to be such that it starts at one of the vertices and draws a side then turns by a certain angle then draws a side again until a full outline of a star is drawn. It needs to be able to compute the angle each time for any side length I may give it. So I'm more looking to understand what the realtionships are between the angles the shape and the side lengths.
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  8. #8
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    Quote Originally Posted by Alyosha View Post
    Draw the figure, or sketch it.
    (I don't know how to draw in computers so I will just describe it)
    Draw it such that one of the 5 "star-points" is the topmost. Call that point, A.
    Draw a vertical line segment from A to the center of the star (or of the circumscribing circle). Call that center, point O.
    Draw a line segment from O to one of the bases (lower ends) of the isosceles triangle (the "star-point"). Call that end or "inside corner of the star" as point B.
    Draw a horizontal line segment connecting the two lower ends of the star-point. Call the intersection of that line segment with AO as point C.

    I'm not sure I understand this. AO runs from the topmost point of the star to the center, if draw a horizontal line segment connecting the two lower ends of the star-point, it doesn't intersect with AO as AO stops in the middle of the star.
    :-)

    The lower ends of the top "star-point" are not the points of the lower two star-points".
    The said "starpoint" is the isosceles triangle whose equal sides are s and s. The angle between those s and s is 36 degrees.

    5-pointed star.
    That's why I called "starpoint" one inverted V.
    There are 5 inverted V's in the whole 5-pointed star.

    Hard to explain. Hard to understand.
    I stop here.
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  9. #9
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    Quote Originally Posted by ticbol View Post
    :-)

    The lower ends of the top "star-point" are not the points of the lower two star-points".
    The said "starpoint" is the isosceles triangle whose equal sides are s and s. The angle between those s and s is 36 degrees.

    5-pointed star.
    That's why I called "starpoint" one inverted V.
    There are 5 inverted V's in the whole 5-pointed star.

    Hard to explain. Hard to understand.
    I stop here.
    Ahh, got it. Excellent, I can do it now.

    Thanks alot.
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  10. #10
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    Quote Originally Posted by Alyosha View Post
    The program needs to be such that it starts at one of the vertices and draws a side then turns by a certain angle then draws a side again until a full outline of a star is drawn. It needs to be able to compute the angle each time for any side length I may give it. So I'm more looking to understand what the realtionships are between the angles the shape and the side lengths.
    Oh, that is all?
    You should have mentioned that in your original question.

    So start anywhere.
    From any point, draw the s going down.
    Then turn 108 degree up.
    Draw another s.
    Then turn 36 degrees down.
    Draw another s.
    Then turn 108 degrees "up". ------"up" means not really up. Just not the same direction of the previous s.
    Draw another s.
    Then turn 36 degrees "down"
    Etc...until you connect to the initial point where you started.
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