A wheel clamp attaches to a vehicle wheel for performing a wheel alignment on the vehicle. The wheel clamp has a body for supporting a target or measuring head, and three extendable arms slidably mounted to the body. Each arm has a gripping portion for gripping the tire's tread surface and a sidewall contact portion for contacting the tire sidewall such that the clamp body is substantially parallel to the vehicle wheel and the wheel clamp has no contact with the rim. A self-centering linkage has enmeshed gears and link arms respectively connecting the arms to the gears, such that when the gears are rotated, the arms simultaneously proportionally slide relative to the clamp body, so the gripping portions of the arms engage and grip the tire tread surface and tighten the clamp onto the tire, while the sidewall contact portions of the arms contact the tire sidewall.