Graphical addition of velocities in Minkowski spacetime In what follows we are in the domain of special relativity.
Consider a reference frame with two velocity vectors $\mathbf{P}$ and $\mathbf{Q}$ with their tails at the origin. How do I represent relativistic combination of these two velocities graphically on a normal sheet of graph paper? 
Let me explain a bit more. Suppose $\mathbf{P}$ and $\mathbf{Q}$ are time-like vectors with length $p$ and $q$ respectively. I choose an origin on the graph paper, call it $O$, and draw two hyperbolas: $t^2-x^2=p^2$ and $t^2-x^2=q^2$. From the origin I draw lines along the known directions of $\mathbf{P}$ and $\mathbf{Q}$, and their intersection with the previously mentioned hyperbolas (and in that order) determine the points $P$ and $Q$. Then I draw the vectors $\vec{OP}$ and $\vec{OQ}$ on the graph paper. Which graphical construction will now give me the sum of these two vectors on the graph paper? What if one of the vectors is space-like or null?
P.S. I have looked at this post but it doesn't answer my question. Searching the internet throws up papers on "gyro-vectors", but so much math seems pointless just to learn how to add vectors on graph paper.
 A: For relativistic velocity addition in the plane, there is a highly elegant graphical construction detailed in:
John A. Rhodes and Mark D. Semon, "*Relativistic velocity space, Wigner rotation, and Thomas precession", Am. J. Phys. 72 #7, July 2004
That is, this construction lets you visualize the composition of any two members of the group $\mathrm{SO}(1,\,2)$. It gives you both the combined velocity and the Wigner rotation (recall that the addition of two non-collinear boosts is not a boost, but rather a boost combined with a rotation, as I discuss further here).
I have built a Mathematica demonstration which draws the graphical construction at:
Rod Vance, "Boost Composition and Wigner Rotation in Rhodes-Semon Rapidity Space", Wolfram Demonstrations Project Published: November 2, 2015
Basically, one represents boosts in the Poincaré disk model of Hyperbolic space: the boost of rapidity $\eta$ making an angle $\phi$ with the $x$ axis is represented by the complex number $z = e^{i\,\phi}\,\tanh\left(\frac{\eta}{2}\right)$. I sketch the construction below:

Our first boost is the ray $\vec{OA}$, and the addition triangle is the hyperbolic triangle $\triangle OAB$. The second boost bears the composite along the circular arc $AB$, where the blue tangent to the arc at the point $A$ is in the direction of the second boost. The arc is uniquely defined by this tangent and the lemma that all such arcs make right angles with the unit circle at the two points where they meet the unit circle. The composed boost is the point $B$. The Wigner rotation is the angle subtended by the arc $AB$ at the circle's center $C$.
If $z_1$ is the first boost, followed by the boost represented by the point $z_2$, then the combined boost's rapidity and direction is represented by the point:
$$z_3 =\frac{z_1+z_2}{z_1+z_2\,z_1^\ast}\tag{1}$$
and the accompanying Wigner rotation is, as stated above, the angle subtended by arc $AB$ at the circle center $C$. The arc $AB$ defines the trajectory of the motion state if a frame initially moving inertially relative to the observer as described by $z_1$ is acted on by a constant acceleration in the direction of the blue tangent.
Of course, the method is not wholly graphical, but then you also need to calculate hyperbolic tangents to find the points $z_1$ and $z_2$ anyway, but it does give you a great deal of intuition for both the change of motion state under constant acceleration. It's actually very like doing impedance transformation calculations with a Smith Chart, which is also a graphical calculator for a Möbius transformation of $z_2$ in the form of equation (1). Here, as in the Smith Chart, you could simply use a nomogram to transform between velocities and rapidities to find the correct points on the chart. In fact, you can use a rotated unmodified Smith Chart for your calculation, because one set of the two orthogonal families on the Smith Chart are precisely geodesics in the Poincaré disk - i.e. circles that meet the unit circle at right angles.
If you are really purist, you can maximize graphical construction one of two ways.


*

*Note that $|z_1+z_2|$, $|z_1|$ and $\left|1+z_2\,\frac{z_1^\ast}{z_1}\right| = \left|1+z_2\,\exp(-2\,i\,\arg(z_1)\right|$ are all readily found by graphical construction given vectors representing $z_1$ and $z_2$, thus the ratio $|z_3|=\left|\frac{z_1+z_2}{z_1+z_2\,z_1^\ast}\right|$ is readily found by real multiplications and divisions. One then simply needs to construct the intersection of the circular acceleration arc and the circle $|z|=|z_3|$ with a compass;

*Begin with the point $z_2$ and then note that it is mapped to the composite boost point $z_3$ by the composition of the following operations in the following order: $z\mapsto \exp(-2\,\arg(z_1))\,z$, $z\mapsto 1+z$, $z\mapsto z^{-1}$, $z\mapsto \left(1-{z_1^\ast}^{-1}\right)\, z$, $z\mapsto z + {z_1^\ast}^{-1}$. There are standard graphical straightedge and compass constructions for all of these involving the Circle of Apollonius and parallelogram construction (where ${z_1^\ast}^{-1}$ and $1-{z_1^\ast}^{-1}$ are also found from $z_1$ by the same constructions).
