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Math Help - Paraboloid Mirror

  1. #1
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    Paraboloid Mirror

    can any one help me ?
    i need to calculate the depth of curve in a elliptical paraboloid mirror.
    the mirror in question is a 200mm long,30mm wide and 30mm deep with a focal length of 3.2m.
    i can workout a spherical depth of curve but this has me beat.
    thanks sagittal
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  2. #2
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    What is "depth of curve"?

    I searched Google, Ask.com, Yahoo, Wikipedia. Nothing there explained to me the meaning of depth of curve.

    Is depth of curve the distance from the focal point to the vertex?
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  3. #3
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    depth of curve

    depth of curve is the lowest point of the mirror(in this case the center of the 200mm)
    depth of curve of a mirror is called the sagitta; it is the difference in height between the edge of the mirror and the center of the mirror.
    the sagitta is determind by the radius of curvature(in this case the focal length of 3.2m = 6.4m ROC)
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  4. #4
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    depth of curve

    Quote Originally Posted by ticbol View Post
    What is "depth of curve"?

    I searched Google, Ask.com, Yahoo, Wikipedia. Nothing there explained to me the meaning of depth of curve.

    Is depth of curve the distance from the focal point to the vertex?
    depth of curve is the lowest point of the mirror(in this case the center of the 200mm)
    depth of curve of a mirror is called the sagitta; it is the difference in height between the edge of the mirror and the center of the mirror.
    the sagitta is determind by the radius of curvature(in this case the focal length of 3.2m = 6.4m ROC)
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  5. #5
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    Quote Originally Posted by sagittal View Post
    depth of curve is the lowest point of the mirror(in this case the center of the 200mm)
    depth of curve of a mirror is called the sagitta; it is the difference in height between the edge of the mirror and the center of the mirror.
    the sagitta is determind by the radius of curvature(in this case the focal length of 3.2m = 6.4m ROC)
    I forgot about this.

    So, that is the depth of curve, (doc).
    Then, if the mirror is given with depth = 30mm, the doc is less than 30mm. So that there is some thickness of mirror material below the doc.

    If the mirror is given as 200mm long and 30mm wide, does that mean the edge of the elliptic paraboloid, whose vertex is in the center of the mirror, passes through the ends of those dimensions passing through the center?
    There are no thickness of the mirror to protect the edges of the top pof the mirror--like at the bottom?

    Let me assume the paraboloid passes through the ends of those dimensions.

    Let us use the widest parabola, the parabola that passes through tthe middle of the width opf the mirror.
    ---With the origin (0,0) at ther center of the bottom of the mirror, the vertex of the widest parabola is at (0,y_0). Let's call y_0 as d. So depth of curve is 30mm minus y_0 = (30 -d) mm
    ---The right edge of the top of the mirror is then point (200/2,30) or (100,30).
    ---The focus is at (0, d mm +3.2 m) or at (0,3200+d).

    One standard form of the equation of parabola is
    (y -k) = [1 /(4p)]*[(x -h)^2] -------------------------(i)
    where
    (h,k) is the vertex
    p is the focal length.
    So, using that, and p=3200mm, and (h,k) = (0,d):
    (y -d) = [1 / (4*3200)][(x -0)^2]
    y -d = (1/12,800)x^2 ------------------------------(ii)

    At point (100,30),
    30 -d = (1/12,800)(100^2)
    30 -d = 0.78125
    d = 30 -0.78125
    d = 29.21875 mm

    Therefore, depth of curve = 30 -29.21875 = 0.78125 mm. -----answer.
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  6. #6
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    sagitta

    i think you maybe wrong.

    a spherical mirror would be 0.00078125 sagitta.
    2 2
    the formula would be sagitta=mirror radius/(4xfocal lenght) r /4f.


    2
    in this case 200mm lenght=radius 0.1 /4x3.2m=0.00078125

    a paraboloid mirror would be a deeper sagitta.
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  7. #7
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    I don't know about the sagitta for sherical mirror.

    My computations are those for the sagitta(?) of the elliptic paraboloid mirror based on the given facts.
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  8. #8
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    paraboloid

    you were right ,
    Technically, they are different (at the wavelength of light scale, hundreds of nanometers).
    Practically, they are the same (at the scale you can measure to, ~0.025 mm)
    thanks sagittal
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