Can inline skates carve an arc?

1 - The Cone Effect:

• The section of wheel on the floor, can be considered as conical in shape.
• As a result, the part which is closer to the bearings has a shorter circumference than the part further from the bearings.
• All parts of the wheel rotate together.
• During a single rotation, the part nearer the bearings travels a shorter distance.
• The only situation in which this occurs is if the wheel traces an arc during this rotation.
• The direction of this arc will be towards the bearings, which in the case of skating means the direction in which the wheel is angled.
• The radius of the arc traced is determined by the difference in the circumference of the 2 extremities of the section of wheel touching the floor.
• To travel in a straight line, either the 'inside' or 'outside' extremity must slip slightly.

   axis      /\  section
      "_    /  \ of wheel
        "_ /   /
          "   /
         /   /   -------->
        /   /    direction
       /___/     of arc
  

2 - The Non-concentric Effect:

• All of the wheels on a skate are in a fixed line.
• The wheels all point along the line connecting them.
• All of the wheels are touching the ground at all times.
• When tracing an arc, a wheel must be point in a direction tangential to the arc at the point of contact.
• Given the first 2 points, none of the wheels can possibly be tangential to the same arc.
• They cannot even be tangential to concentric arcs, so they must rotate around different points.
• If all of the wheels are to trace arcs around the same centre, only 1 wheel can be exactly tangential to this centre, the others cannot be.
• If they are forced into this arc, they will need to slip slightly.

     .                                     .
      ".                                 ."
         "-.                         .-"    arc being traced
             "-.__             __.-"
                  ###-.###.-###  ###
                   1    2    3    4
  

• Only wheel 2 is pointing along the line of the arc. The others are pointing slightly away from the arc, so will be 'dragged' around the arc.

So, for a wheel at an angle to travel in a straight line, some slippage must occur, but wheels pointing in the same direction cannot follow an arc without slippage.

Either way there is some slippage. Therefore the least slippage will probably occur when the skate traces an arc somewhere between the thoeretical one defined by method 1, and the straight line.

This does not prove that it is impossible to trace an arc on one skate, just that some slippage must occur if more than 1 wheel is touching the ground at any one time. Of course, whith a rockered setup, it is quite possible that only 1 wheel is on the ground, making 'The Non-concentric Effect' invalid.

Alternatively, other effects may help the skate to turn, including

1. Wheel/frame deformation - has 2 effects, the wheels can turn to point parallel to the arc, and the contact point on the wheel can move up, putting it closer to the arc.
2. Wheel slip - When the wheel is not on the arc, or is not pointing along the arc (but still on the floor) it must slip to some extent to follow the arc.
3. Wheel lift - If one or more wheels is not touching the floor, then this wheel will not restrict the turning of the skate.
4. Bearing tolerences - The movements on the bearings will have the same effects as wheel and frame deformation.
5. Specially shaped wheels - If, when gliding the center wheels contact the ground lower down than the front-back wheels. This creates an arc similar to an ice-skate by bringing the contact points of the wheels closer to the arc being traced.