  The strange visual appearance of objects is one of the puzzling predictions
of Einstein's relativity. This is mainly due to the distinction between
measuring and seeing, where the former is described by the Lorentz
Transformation and the latter considers the time light rays (emitted by each
point on the object) take to reach the observer. We compute the apparent
position of a point given its velocity, initial position, and observation time.
The apparent speed of a point is calculated, and we obtain that it exceeds the
speed of light when approaching the observer, similar to superluminal motion.
For parameterizable surfaces, we analyze properties (such as curvature and
torsion) of apparent shapes. The observation that a sphere retains its circular
silhouette when transformed to its apparent shape, independent of the initial
conditions, is proved mathematically. Plots describing the apparent speed and
length of objects are made, and the metric tensor for a distorted sphere is
calculated. A generalized equation for the Doppler effect and relativistic
aberration is derived to analyze regions of redshift and blueshift. Using the
Born-rigidity conditions, we compute the hyperbolic trajectories of each point
on an extended object given an initial velocity, position, and proper
acceleration for any reference point. The claim that a rigid body, accelerating
in Special Relativity, cannot exceed a given length in certain circumstances is
justified. We obtain many non-trivial results, which are proved algebraically
and using light cones, that are tested by taking the limit of acceleration
approaching 0 to retrieve results in the constant velocity scenario. In
conclusion, these visualizations may be used by teachers to explain SR
intuitively. Finally, we provide an overview of extending the same problem to
curved spacetime and explain the potential applications of this project.
