Vector direction

I would have attached the document (with pretty pictures) I have but form some reason I am unable to attach or link anything. This document helped me understand a whole lot more about vectors a couple of years ago. Feel free to PM me with your email address if you want the actual document.

Direction Vectors – Programmable CMMs Push the Envelope
By Richard Clark

This story was originally published in a 2003 issue of Tooling and Production magazine. It can be viewed on the World Wide Web at the following link: http://www.manufacturingcenter.com/tooling/archives/0203/0203qmvector.aspVectors

There is no more exciting measurement system gaining momentum today than CNC CMMs. More and more facilities are realizing how much time is saved by the ability to measure a part while recording a PPG (part program) and then measuring the next 50-100 parts with the click of a mouse. The true phenomenon involved with these machines is not how fast they measure, but rather how precise they can perform as compared to their predecessor, the manual CMM. This precision can be explained in 2 words – Direction Vectors.

Many of today’s versions of CNC CMM software have the capability to use a
Teaching mode where circles, planes, and cylinders can be measured in an auto-measure mode. This teaching mode actually writes the program from information given about the feature (Diameter, Depth. Location, etc.). Because of variables such as (but not limited to) the configuration of the part or part fixture device preventing a complete 360° measurement, some features cannot be programmed in this manner. The operator must manually type in different types of movements to program the feature for the first time. This is where an understanding how vectors work is extremely important.

Direction vectors are calculated using a very basic Trigonometry concept called the unit circle diagram. Imagine a ring gage placed on a CMM for a reference measurement. The top surface of the ring can be used as the XY baseplane, Z-axis origin. The center of the ring gage is the X, Y origin.



Vectors can be calculated in I-J-K or L-M-N formats. Once you’ve learned how they work, L-M-N can be self-taught in 2 minutes.

How many points you wish to probe determines the next step. To make it simple we’ll use 18 points. This breaks down to a point taken every 20° of the circle. Think of our circle in the Polar Coordinate System of Radius, Angle, and Height with the X positive line serving as 0°.


In our 18-point circle, a point will be taken every 20° in a counter-clockwise rotation. This is where the unit circle diagram takes over.
The diagram works by using the 20° angle and creating a right triangle within the circle as shown below. Since the right angle is 90° and we created a 20° angle, the last angle will be 70°. This diagram and a $10 calculator are all you need to calculate vectors.


I-J-K vectors are the cosine values of the X, Y, and Z angles. The angle from the X axis is 20° which has a cosine of 0.93969, the angle from the Y axis is 70°, which has a cosine of 0.34202, the Z axis is 0° (or some purists view this as 90°), both of which have a cosine of 0. We now have our vector direction for the point:

I - 0.93969
J - 0.34202
K - 0

In the Spatial format, 3 the angles themselves describe the direction vector


L - 20°
M - 70°
N - 90°

The direction of the probe follows this path:



Our next point will be taken after a 40° CCW rotation within the circle. When we form this right triangle within the circle it consists of a 40° angle from the X-axis and a 50° angle from the Y-axis.



When we calculate the cosines, it looks like this:
I - 0.76604
J - 0.64279
K - 0
The spatial version would become:

L – 40°
M – 50°
N – 90°

The direction of the probe follows this path:



And that’s how direction vectors work. It is a textbook example of how the most technical examples of Metrology are nothing more than the same basic deck of Mathematical “playing cards” shuffled a little differently. If you learn to deal this game you’ll certainly have your Programmable CMM moving in the right direction… pun intended.

Richard Clark works as a Metrology Consultant and CMM operator in Portland Indiana, to receive a freeware version of his “Vector Direction Calculator” e-mail feedback to rcmetrology@yahoo.com








Vectors – The True Direction of Programmable CMMs

By Richard Clark

This story is scheduled for publication in an upcoming issue of CMM Monthly magazine. To subscribe to CMM Monthly (free of charge) and/or download past issues in Adobe format, visit them on the World Wide Web at https://www.cmmmonthly.com

If you ask any extremely dedicated and highly motivated DCC (Direct Computer Controlled) CMM programmer what is the foundation of a programmable CMM's capability and precision and they will answer with 2 words – Direction Vectors.

The 1st step to getting direction vectors "figured out" is to understand their origin is a very basic Trigonometry concept called the unit circle diagram. We can probably all remember the phrase - Sine, Cosine, and Tangent; Oscar had a heap of apples. If you have the time you can brush up on the 3 main trig functions but all you really need to calculate vectors is a $3 calculator with a cosine function.

Direction vectors are normally expressed in an I-J-K format.

I - The cosine of the angle shift or rotation from the X-axis
J - The cosine of the angle shift or rotation from the Y-axis
K - The cosine of the angle shift or rotation from the Z-axis

A correct vector will command a probe movement direction that is perpendicular to the surface being measured.

The best example to learn the calculation with is a circle measurement. Imagine a ring gage with the X, Y origin established at the center of the ring gage inside diameter.


If we wanted to create a 10-point circle measurement program, a point would be taken every 36° "around" the circle.

This is where the Unit Circle Diagram works by plotting the 1st point at 36° and creating a right triangle within our ring gage.



The angle rotation from the X-axis is 36° which has a cosine value of 0.80902, the angle rotation from the Y-axis is 54°, which has a cosine value of 0.58779, the Z axis angle is 0°, which has a cosine of 0. We now have our vector direction for the point:

I = 0.80902
J = 0.58779
K= 0

The probe follows the path of the vector direction:


To further illustrate how simple this can be we will break the process down into 3 steps and use some fundamental Geometry principles that you’ve probably been familiar with for years.

In the last example we started with a 36° angle that was a feature within a right triangle projected to the XY plane.

Step #1 – Identify 1 known angle and determine which axis the angle rotates from. Our known angle is 36° and it rotates from the X-axis.

Step #2 – Calculate the unknown angle using the “90 minus rule” and determine the rotation from which axis. Since we are dealing with a right triangle and we already have a 36° angle, the unknown angle (90-36) = 54° and it rotates from the Y-axis.

Step #3 – Calculate the I-J-K vectors using the Cosine of the angles.
I = 0.80902 (The Cosine of our 36° X angle)
J = 0.58779 (The Cosine of our 54° Y angle)
K= 0 (Our probe is not moving along the Z axis)

Calculating the correct vector for probing points located on a line feature may seem to be more in-depth but they are just as simple.

In our next example we wish to calculate the direction vector to probe a point located on a feature that is neither parallel nor perpendicular to the X or Y-axes. We can determine from our print specifications that a 210° angle would be perpendicular to the surface of being measured.


This is clearly illustrated when the Part Coordinate System is viewed using Polar coordinates.



Step #1 – We have a known angle of 210° that rotates from the X-axis.

Step #2 – Using the “90 minus rule” we can determine the unknown angle to be -120° and since we are in the XY plane the angle rotation is from the Y-axis.

Step #3 – Calculate the I-J-K vectors using the Cosine of the angles.
I = -0.86603 (The Cosine of our 210° X angle)
J = -0.50000 (The Cosine of our -120° Y angle)
K= 0 (Our probe is not moving along the Z axis)


What a concept! The mystery and magic of direction vectors can be revealed using only 1 angle and 1 trigonometry function. This is nothing more than a simple proof that some of the most seemingly difficult measurement systems can be deciphered using the one and only true universal language: Mathematics.


Richard Clark works as a Metrology Consultant and CMM operator in Portland Indiana, to receive a freeware version of his “Vector Direction Calculator 4.02” e-mail feedback to rcmetrology@yahoo.com

I would have attached the document (with pretty pictures) I have but form some reason I am unable to attach or link anything. This document helped me understand a whole lot more about vectors a couple of years ago. Feel free to PM me with your email address if you want the actual document.

Direction Vectors – Programmable CMMs Push the Envelope
By Richard Clark

This story was originally published in a 2003 issue of Tooling and Production magazine. It can be viewed on the World Wide Web at the following link: http://www.manufacturingcenter.com/tooling/archives/0203/0203qmvector.aspVectors

There is no more exciting measurement system gaining momentum today than CNC CMMs. More and more facilities are realizing how much time is saved by the ability to measure a part while recording a PPG (part program) and then measuring the next 50-100 parts with the click of a mouse. The true phenomenon involved with these machines is not how fast they measure, but rather how precise they can perform as compared to their predecessor, the manual CMM. This precision can be explained in 2 words – Direction Vectors.

Many of today’s versions of CNC CMM software have the capability to use a
Teaching mode where circles, planes, and cylinders can be measured in an auto-measure mode. This teaching mode actually writes the program from information given about the feature (Diameter, Depth. Location, etc.). Because of variables such as (but not limited to) the configuration of the part or part fixture device preventing a complete 360° measurement, some features cannot be programmed in this manner. The operator must manually type in different types of movements to program the feature for the first time. This is where an understanding how vectors work is extremely important.

Direction vectors are calculated using a very basic Trigonometry concept called the unit circle diagram. Imagine a ring gage placed on a CMM for a reference measurement. The top surface of the ring can be used as the XY baseplane, Z-axis origin. The center of the ring gage is the X, Y origin.



Vectors can be calculated in I-J-K or L-M-N formats. Once you’ve learned how they work, L-M-N can be self-taught in 2 minutes.

How many points you wish to probe determines the next step. To make it simple we’ll use 18 points. This breaks down to a point taken every 20° of the circle. Think of our circle in the Polar Coordinate System of Radius, Angle, and Height with the X positive line serving as 0°.


In our 18-point circle, a point will be taken every 20° in a counter-clockwise rotation. This is where the unit circle diagram takes over.
The diagram works by using the 20° angle and creating a right triangle within the circle as shown below. Since the right angle is 90° and we created a 20° angle, the last angle will be 70°. This diagram and a $10 calculator are all you need to calculate vectors.


I-J-K vectors are the cosine values of the X, Y, and Z angles. The angle from the X axis is 20° which has a cosine of 0.93969, the angle from the Y axis is 70°, which has a cosine of 0.34202, the Z axis is 0° (or some purists view this as 90°), both of which have a cosine of 0. We now have our vector direction for the point:

I - 0.93969
J - 0.34202
K - 0

In the Spatial format, 3 the angles themselves describe the direction vector


L - 20°
M - 70°
N - 90°

The direction of the probe follows this path:



Our next point will be taken after a 40° CCW rotation within the circle. When we form this right triangle within the circle it consists of a 40° angle from the X-axis and a 50° angle from the Y-axis.



When we calculate the cosines, it looks like this:
I - 0.76604
J - 0.64279
K - 0
The spatial version would become:

L – 40°
M – 50°
N – 90°

The direction of the probe follows this path:



And that’s how direction vectors work. It is a textbook example of how the most technical examples of Metrology are nothing more than the same basic deck of Mathematical “playing cards” shuffled a little differently. If you learn to deal this game you’ll certainly have your Programmable CMM moving in the right direction… pun intended.

Richard Clark works as a Metrology Consultant and CMM operator in Portland Indiana, to receive a freeware version of his “Vector Direction Calculator” e-mail feedback to rcmetrology@yahoo.com








Vectors – The True Direction of Programmable CMMs

By Richard Clark

This story is scheduled for publication in an upcoming issue of CMM Monthly magazine. To subscribe to CMM Monthly (free of charge) and/or download past issues in Adobe format, visit them on the World Wide Web at https://www.cmmmonthly.com

If you ask any extremely dedicated and highly motivated DCC (Direct Computer Controlled) CMM programmer what is the foundation of a programmable CMM's capability and precision and they will answer with 2 words – Direction Vectors.

The 1st step to getting direction vectors "figured out" is to understand their origin is a very basic Trigonometry concept called the unit circle diagram. We can probably all remember the phrase - Sine, Cosine, and Tangent; Oscar had a heap of apples. If you have the time you can brush up on the 3 main trig functions but all you really need to calculate vectors is a $3 calculator with a cosine function.

Direction vectors are normally expressed in an I-J-K format.

I - The cosine of the angle shift or rotation from the X-axis
J - The cosine of the angle shift or rotation from the Y-axis
K - The cosine of the angle shift or rotation from the Z-axis

A correct vector will command a probe movement direction that is perpendicular to the surface being measured.

The best example to learn the calculation with is a circle measurement. Imagine a ring gage with the X, Y origin established at the center of the ring gage inside diameter.


If we wanted to create a 10-point circle measurement program, a point would be taken every 36° "around" the circle.

This is where the Unit Circle Diagram works by plotting the 1st point at 36° and creating a right triangle within our ring gage.



The angle rotation from the X-axis is 36° which has a cosine value of 0.80902, the angle rotation from the Y-axis is 54°, which has a cosine value of 0.58779, the Z axis angle is 0°, which has a cosine of 0. We now have our vector direction for the point:

I = 0.80902
J = 0.58779
K= 0

The probe follows the path of the vector direction:


To further illustrate how simple this can be we will break the process down into 3 steps and use some fundamental Geometry principles that you’ve probably been familiar with for years.

In the last example we started with a 36° angle that was a feature within a right triangle projected to the XY plane.

Step #1 – Identify 1 known angle and determine which axis the angle rotates from. Our known angle is 36° and it rotates from the X-axis.

Step #2 – Calculate the unknown angle using the “90 minus rule” and determine the rotation from which axis. Since we are dealing with a right triangle and we already have a 36° angle, the unknown angle (90-36) = 54° and it rotates from the Y-axis.

Step #3 – Calculate the I-J-K vectors using the Cosine of the angles.
I = 0.80902 (The Cosine of our 36° X angle)
J = 0.58779 (The Cosine of our 54° Y angle)
K= 0 (Our probe is not moving along the Z axis)

Calculating the correct vector for probing points located on a line feature may seem to be more in-depth but they are just as simple.

In our next example we wish to calculate the direction vector to probe a point located on a feature that is neither parallel nor perpendicular to the X or Y-axes. We can determine from our print specifications that a 210° angle would be perpendicular to the surface of being measured.


This is clearly illustrated when the Part Coordinate System is viewed using Polar coordinates.



Step #1 – We have a known angle of 210° that rotates from the X-axis.

Step #2 – Using the “90 minus rule” we can determine the unknown angle to be -120° and since we are in the XY plane the angle rotation is from the Y-axis.

Step #3 – Calculate the I-J-K vectors using the Cosine of the angles.
I = -0.86603 (The Cosine of our 210° X angle)
J = -0.50000 (The Cosine of our -120° Y angle)
K= 0 (Our probe is not moving along the Z axis)


What a concept! The mystery and magic of direction vectors can be revealed using only 1 angle and 1 trigonometry function. This is nothing more than a simple proof that some of the most seemingly difficult measurement systems can be deciphered using the one and only true universal language: Mathematics.


Richard Clark works as a Metrology Consultant and CMM operator in Portland Indiana, to receive a freeware version of his “Vector Direction Calculator 4.02” e-mail feedback to rcmetrology@yahoo.com