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Helix in architecture: Spiral Stairs

Helix in architecture: Spiral Stairs

"Spiral" Stairs are stairs that go up together with tunling around a central axis.

They follow a mathematical curve called "helix".

But what is the equation of an helix?

Let's suppose that the vertical axis "z" is the central axis around wich the helix turns, oriented to the top, and that the first horizontal axis "x" is along the basis of the stairs.

Let the parameter a be the horizontal angle of a dot on the curve with the "x" axis.  Then the horizontal projection of the dot has coordinates x=Rcos(a) and y=-Rsin(a) if its turns clockwise or Rsin(a) otherwise, where R is the distance of the spiral with the vertical axis "z".

Note that a may be in degrees or radians, so that the cosine and sinus functions are adapted to the unit of a.

Let us now calculate the "z" coordinate.

As the helix goes up when turning, z is an increasing function of the angle a.

As a matter of fact, it is a linear function of a: z=Ca, where Z=C/360° (or C divided by 2 pi if a in Radians) is the height during which the stairs make a complete turnaround.

Thus the equation of the helix as a function of the height z by which the dot goes up is
x=Rcos(z/C) and y=(-)Rsin(z/C), z/C modulo 360° or 2pi being the horizontal angle of the radius with the "x" axis.

Now, to calculate the positions of the individual steps, let us suppose that:

  • The stairs make a complete turnaround for an elevation of Z=2.50 meters (a floor)
  • There are N=20 steps per floor

Then the height by which the stairs go up for 1 step is Z/N=2.50/20=0.125 m=125 mm.

The angle in degrees by which the stairs turn around the central axis for 1 step is
da=360°/N=360°/20=18°, so that the coordinates of the successive steps on "x" and "y" axis are, for k=0 to the total number of steps -1:

  • xk = cos(k x 18°)
  • yk = (-)sin(k x 18°) (with "-" for clockwise turning stairs)

More precisely, the xk and yk are the coordinates of the beginning of the steps and we should take into account the wifth of each step. But these are the beginnig of the next step for each step, so it is already computed.

Now, you are ready to design spiral or helical stairs like a guenuine architect!


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