Hyperbolic Paraboloids – math model
Wednesday, November 11th, 2009Hyperbolic paraboloids (HP) are discribed by the formula
The below script creates HPs in Microstation within a certain range of x and y coordinates. The dashed lines indicate the regular grid on coordinate system. These are of parabolic shape. The thick diagonal lines are all straight. They are the reason why this shape is interesting for construction, as it can be discribed as a series of straight lines. Intersecting the surface with a horizontal plane creates hyperbolas (blue curves).
Hyperbolic paraboloid
‘Hyperbolic Paraboloids
‘Sebastian Gmelin
’11.11.2009
Const a As Double = 2
Const b As Double = 2
Const range As Long = 8
Dim hp() As Point3d
Dim x As Double
Dim y As Double
Dim z As Double
Dim point As Point3d
Dim ele As element
Sub drawGrid()
Dim points() As Point3d
ReDim points(range * 2)
For x = range * -1 To range
For y = range * -1 To range
points(y + range) = hp(x + range, y + range)
Next y
Set ele = CreateLineElement1(Nothing, points)
ele.Color = 4
ele.LineWeight = 0
ActiveModelReference.AddElement ele
ele.Redraw
Next x
For y = range * -1 To range
For x = range * -1 To range
points(x + range) = hp(x + range, y + range)
Next x
Set ele = CreateLineElement1(Nothing, points)
ele.Color = 4
ele.LineWeight = 0
ActiveModelReference.AddElement ele
ele.Redraw
Next y
End Sub
Sub drawDiaGrid()
Dim points() As Point3d
Dim i As Long
For i = 1 To range * 2
x = range * -1
ReDim points(i)
For y = range – i To range
points(y – range + i) = hp(x + y + i, y + range)
Next y
Set ele = CreateLineElement1(Nothing, points)
ele.Color = 7
ele.LineWeight = 1
ActiveModelReference.AddElement ele
ele.Redraw
For y = i To 0 Step -1
points(i – y) = hp(i – y, y)
Next y
Set ele = CreateLineElement1(Nothing, points)
ele.Color = 7
ele.LineWeight = 1
ActiveModelReference.AddElement ele
ele.Redraw
If i < range * 2 Then
y = range * -1
ReDim points(range * 2 – i)
For x = range * -1 + i To range
points(x + range – i) = hp(x + range, y + range * 2 + x – i)
Next x
Set ele = CreateLineElement1(Nothing, points)
ele.Color = 7
ele.LineWeight = 1
ActiveModelReference.AddElement ele
ele.Redraw
y = range
ReDim points(range * 2 – i)
For x = range * -1 + i To range
points(x + range – i) = hp(x + range, y – x + i)
Next x
Set ele = CreateLineElement1(Nothing, points)
ele.Color = 7
ele.LineWeight = 1
ActiveModelReference.AddElement ele
ele.Redraw
End If
Next i
End Sub
Sub horizontalCut(cz As Double)
Dim points() As Point3d
Dim pointsNeg() As Point3d
ReDim points(range * 2)
ReDim pointsNeg(range * 2)
If cz >= 0 Then
For y = range * -1 To range
x = Math.Sqr(a * a * (((y * y) / (b * b)) + 2 * cz))
points(y + range) = Point3dFromXYZ(x, y, cz)
pointsNeg(y + range) = Point3dFromXYZ(-1 * x, y, cz)
Next y
Else
For x = range * -1 To range
y = Math.Sqr(b * b * (((x * x) / (a * a)) – 2 * cz))
points(x + range) = Point3dFromXYZ(x, y, cz)
pointsNeg(x + range) = Point3dFromXYZ(x, -1 * y, cz)
Next x
End If
Set ele = CreateLineElement1(Nothing, points)
ele.Color = 5
ele.LineWeight = 2
ActiveModelReference.AddElement ele
ele.Redraw
Set ele = CreateLineElement1(Nothing, pointsNeg)
ele.Color = 5
ele.LineWeight = 2
ActiveModelReference.AddElement ele
ele.Redraw
End Sub
Sub Main()
ReDim hp(range * 2, range * 2)
For x = range * -1 To range
For y = range * -1 To range
z = (((x * x) / (a * a)) – ((y * y) / (b * b))) / 2
point = Point3dFromXYZ(x, y, z)
Set ele = CreateLineElement2(Nothing, point, point)
ele.Color = 3
ele.LineWeight = 10
ActiveModelReference.AddElement ele
ele.Redraw
hp(x + range, y + range) = point
Next y
Next x
drawGrid
drawDiaGrid
horizontalCut (range / 2)
horizontalCut (range / -2)
End Sub
