Putting the Curve into Spacetime

We all know the Lorentz Transformation in its original equation but what if there are other equations that can explain it better and graph it so that the x points are the velocities and the y points are the Lorentz Transformation. Furthermore, what if we can use these new equations to graph time dilation and length contraction equations, as well. And yet, there is another equation that can map out on a graph any particular velocity path as its own graph showing the Lorentz Transformation. This book has unlock these once "hidden" equations from our eyes and may in fact unlock another chapter for the Lorentz Transformation. Plus, I have graphed the Velocity-Additional Formula different than anything I have seen before using the techniques that help me transformed the Lorentz Transformation.