The oval shape of Four Points by Sheraton St. Louis Downtown (Millennium Hotel) is not an ellipse. The South Tower is a composition of arcs that join on common tangent lines. The reason for the common tangent line is to have the arcs join in a visually smooth curve. At P35, P36, P37 and P38 the first derivative of the different arc curves match. The second derivatives do not match since the arcs have different radii at these points where they join.

We set out to find an equation of each arc and use inequalities or absolute value inequalities to specify the x and y values used in that arc.

The red line from P8 to P36 is the radius of the arc between P36 and P37.

The center point is P8 (340.0209,-48.92).

To get the arc for points P36 to P37 we used the equation of a circle:

(x - 340.00209)² + (y + 48.92)² = 23,651.548.

The arc radius is the square root of 23,651.548 which is 153.7906777.

For P36 = (479.4045, 16.07065) to P37 = (479.4045, -113.9107) points in the arc equation above we need to take y values such that:

-113.9107 < y < 16.07065

Do the same process for the arc with center P9 between P35 and P38 to get:

p9 was our point

-113.9107 < y < 16.07065

the arc points were P35 and P38

the distance between P35 and P9 was 153.790768

here is the equation:

(x - 489.9792)² + (y + 48.92)² = 153.790768² = 23651.6003

Then take center point P7 for arc between P35 and P36 and calculate:

We used P7 as the center point to find the distance to P35 & P36. The equation, assuming we chose a point (x,y) that is on the arc being measured, is the square root of (x-x7)² + (y-y7)² = r². Where r = the distance of the arc's radius.

(x - 415)² + (y + 13.96167)² = 71.06251²

350.5955 < x < 479.4045

So with center P6 for arc through P37 and P38 we have:

The center point that on the circle is P6.

From P6 to P37, the distance was 71.062527.

From P6 to P38, the distance squared is also 71.062527.

Then the formula for this arc of the circle:

(x - 415)² + (y + 83.87833)² = 71.062527²

where 350.5955 < x < 479.4045

Student _________
I thought the Millennium hotel was a fascinating structure. I had no idea all the details that went into the planning of this hotel. I liked the part of the building that resembled airplane wings. I may have to go eat dinner there and show people what a neat building it is. Thanks for going "around" there.

Student _________
I loved the experience I had at the Millennium hotel. It was beautiful. I especially liked going up to the 27th floor to the restaurant where Danielle and I were greeted by a lovely African American Hostess who was very nice. She welcomed us to go into the restaurant to experience the movement. We watched the dinning area as it moved and still couldn't believe it. However, we thought that the whole top floor moved, so we were kind of surprised when we found that it didn't. Well, we had fun walking in the dinning area in the opposite direction while it was moving.

Student _________
I had no idea that only a section of the floor in the restaurant revolved. When I heard that it was called the "revolving restaurant", I expected the whole top level to move. When I went up to see the revolving wonder, I realized that only a section of the floor moves. It was very cool. When I stepped on the moving floor I was delighted. It moved very slowly, but still made me feel a little dizzy. I don't know if I would be able to sit down and eat dinner while I'm moving, but I would sure like to try it sometime.

Student _________
The Millennium Hotel really did amaze me. I liked seeing different types of arcs and circles in the Hotel’s architecture and it had many different themes in it's structure. I would find it interesting to know how long the planning and took before the actual structure was built? Another very interesting aspect was the revolving restaurant, "Top of the Riverfront" located at the top floor of the Millennium Hotel- St. Louis. The revolving restaurant surprised me because when you are standing in the restaurant moving, you are not just moving straight but in a circular motion and as you move around and sit and eat dinner, you can see some of St. Louis' most famous landmarks all in 1½ hour meal sitting. I encourage everyone who has never visited the Millennium Hotel in St. Louis to do so! It is an experience worth taking!

Student _________
The Millennium Hotel was pretty cool. It really is an architectural marvel. It was neat how even the indoor pool fit into the design. It was ingenious of the architects to cut the work in half by using symmetry. I have had fun there on a couple of occasions for formal events, but we saw parts of the hotel I had not seen before and that made it more interesting.