Techniques are described for moving an orthodontic object (e.g., an orthodontic appliance and/or its associated tooth) in response to input indicative of a desired movement of the object along a virtual archwire. The movement of objects along the archwire may be used to generate a three-dimensional (3D) representation of a patient's dental arch. The manner in which the object is to be moved along the virtual archwire may be determined automatically during creation of a treatment plan for a patient, or may be determined and input by an orthodontic practitioner as part of the creation of the treatment plan.