## Monday, August 27, 2018

### Rotation maths

My previous post was designed to give you some intuition when working with quaternions. If you already have a library that does multiplications and inverses for you, you should be fine. If you'd like to learn more details or don't have a quaternion library, this blog post is for you.

How do we rotate something by some axis $V=(v_x, v_y, v_z)$ and some angle $\theta_V$?

To answer that, lets start with something simpler first: we ball with a stick in it, the stick is pointing in direction $D=(d_x, d_y, d_z)$. We want to rotate the ball via $V$ to get a new stick direction $R=(r_x, r_y, r_z)$.

We can do this by using Rodrigues' rotation formula.

$R = cos(\theta) D + sin(\theta) (V \times D) + (1-cos(\theta))(V \cdot D) V$

where $\times$ is the cross product (returns a vector)

$(V \times D)_x = (v_y d_z - v_z d_y)$
$(V \times D)_y = (v_z d_x - v_x d_z)$
$(V \times D)_z = (v_x d_y - v_y d_x)$

and $\cdot$ is the dot product (returns a single value)

$(V \cdot D) = v_xd_x + v_yd_y + v_zd_z$

This formula comes from splitting D into two pieces: $D_{\Vert}$, the part that is parallel to V, and $D_{\perp}$, the part that is perpendicular to V. Assuming $V$ and $D$ are normalized,  $D_{\Vert}=(V \cdot D)V$ is the vector projection of $D$ onto $V$, and then $D_{\perp}=D-D_{\Vert}$ (also known as the vector rejection of $D$ onto $V$).

Then we can find $R_{\Vert}$ and $R_{\perp}$ and get $R=R_{\Vert}+R_{\perp}$. (this isn't quite right, there are weighting terms on $R_{\Vert}$ and $R_{\perp}$ based on $\theta_V$, but this is a good intuition)

$D_{\Vert}$ is unaffected by the rotation, so $R_{\Vert}=D_{\Vert}$.

$D_{\perp}$ is rotated $\theta_V$ degrees around $V$. If $\theta_V=90$, $R_{perp}$ will be perpendicular to $D_{\perp}$ and $V$. The cross product of any two vectors is known to return a vector that is perpendicular to them both, so if $\theta_V=90$ degrees, $R_{\perp} = V \times D_{\perp}$. Likewise, if $\theta_V=-90$ degrees, $R_{\perp}= V \times D_{\perp}$, and if $\theta_V=180$ degrees then $R_{\perp}=-D_{\perp}$.

The math for getting $R_{\perp}$ is then identical to what happens when you rotate the 2D vector $(1,0)$ by $\theta_V$ degrees around the origin, just replace $(1,0)$ with $D_{\perp}$ and $(0,1)$ with $V \times D_{\perp}$. See the Rodrigues' Rotation Formula wiki page for the full details.

TODO:

If I have axis angle $V=(v_x, v_y, v_z), v_{\theta}$ and another axis angle $U=(u_x, u_y, u_z), u_{\theta}$, intuitively how we combine them is easy: we apply V to an object, then apply U to an object. But can we make a new axis angle $W=(w_x, w_y, w_z), w_{\theta}$ so that when we apply $W$, it is the same as applying $V$ and then applying $U$?

#### 5 comments:

1. Sega Mega Drive and Genesis Classics | The King of
Sega Mega Drive and Genesis Classics (New) · kirill-kondrashin Alex Kidd in the Enchanted 카지노 사이트 Castle · Alien Soldier · Alien Storm.

2. Many designers choose aluminum sheets because of|as a end result of} they are light-weight, due to this fact simple to handle and set up. While heavier, chrome steel is a superb selection in restaurants or medical amenities, the place sanitation is necessary. Regardless of your specific selection, steel sheets are sometimes assertion items that show an out-of-the-box strategy to designing an interior. In addition, metals sheets can out there in} laser-cut, perforated or corrugated variations, best oversized sweaters which makes them higly customizable and a major raw materials for the inventive designer.

3. In addition to automating the machining process and enhancing precision, CNC eliminates guide tasks and frees machinists to supervise a number of} machines Holders for Toilet Plungers working at the same time. CNC machining is any process the place a part is manufactured by a machine that's controlled by a computer. In milling, a slicing tool spins at high speeds and advances into or throughout the work piece to remove material. Other kinds of CNC machining include turning, routing, plasma slicing, and water jetting. Turning and milling could be mixed into a single machine, as can additive and subtractive operations, creating even more complicated kinds of CNC machining. The CNC system of a machine tool contains the control unit itself, properly as|in addition to} the motion-control system such as the servomotors, drives and axis positioning devices.

4. Shapeways is a distinguished on-line 3D printing service based back in 2007 by Dutch entrepreneur Peter Weijmarshausen, and is now primarily based Women’s House Shoes in New York. However, Shapeways is not just a service but operates as a whole 3D printing marketplace, allowing 3D designers to sell the designs they create on various 3D software program and obtain a reduce of the revenue. Shapeways then fulfills these orders by 3D printing the design and sending it to the buyer.

5. But the corporate has important advantages in an trade that is poised to take off. While it's a sophisticated maneuver, the move into sports activities betting might find yourself enhancing Disney's bottom line. The term “dutching” is often used to describe a betting method that guarantees profit, benefit of|benefiting from|profiting from} free bets and requiring more than one sportsbook 코인카지노 account.