The phrase and the term “orthogonal” are widely used in engineering, but are also often misunderstood.

The other day I was talking to a non-technical friend and I said something like, “Oh, this isn’t going to happen. These two things are orthogonal. ”My friend looked at me and said“ huh? ”And he was right to do so.

Why? Since the term “orthogonal” (from the Latin orthogonus or “rectangular”) and the concept behind it is used in many engineering disciplines, but it is not an everyday term for conversation. The same caution applies to his brother’s term “squaring,” also widely used in engineering, especially in signal theory, signal processing, and communication coding and channels.

This article will look at various aspects of orthogonality and its implications for engineering analysis. We’ll start with an almost intuitive mechanical orthogonal situation, look at the math behind the meaning of an orthogonal, and end with the electronic version, often called “squaring.”

Is “Orthogonal” the same as “uncorrelated”? The answer is “yes, no and maybe” and there are many points of view beyond what we can cover here. In short, “orthogonal” is a mathematical relationship between a continuous signal or process, while “uncorrelated” is a statistical relationship. But the story is more complicated and confusing than that; on References link to some interesting discussions on the subject. However, in general conversation, the two terms are often somewhat synonymous.

Start with mechanical forces

One of the first lessons in basic mechanical engineering is that an arbitrary force at an angle can be divided into two orthogonal components – meaning that they are at right angles to each other – to a selected reference system. (If the system has gravity pulling down, as is often the case, the frame of reference is usually aligned with horizontal and vertical directions. It’s easiest to start with a mechanical example, as it’s somewhat intuitive and familiar to most people).

Consider a force of 400 Newtons (N) exerted at an angle of 60⁰ to move a wagon east (right) on a railway line (Figure 1). The force applied to the vehicle has both an up / down (south) and a left / right component (east) with respect to the selected reference frame. Of course, this is a two-dimensional example; in the real world we have three axes – x, y and z at right angles to each other. Using two dimensions simplifies the discussion and makes the examples easier to follow.

Figure 1: Using basic trigonometry, angular force can be divided into two components at right angles. (Image: Physics classroom)

The sine and cosine functions are used to determine the magnitudes of these two components, along with a basic diagram with a marked angle and a marked hypotenuse. The sine function calculates the component to the south, and the cosine function determines the component to the east.

These are all pretty basic things. But there is still some idea of ​​this: no force in the up / down direction can affect the force in the left / right direction and vice versa. This is because the sine of 0⁰ is zero, as is the cosine of 90⁰.

This independence between the permitted forces is of very practical importance. Suppose you have a rope or wire stretched between two supports, and you want to pull the wire stretched and make it an absolute, perfectly horizontal, quite common goal. To facilitate the analysis, consider the rope weightless. However, there is a weight attached and suspended from its center (if the suspended weight is much heavier than the rope, this is a very valid simplification) (Figure 2). So how hard do you have to pull the rope horizontally to make it go really, absolutely straight?

Figure 2: Can only horizontal force provide the vertical force needed to lift weights? (Image: Question Bank)

The answer is simple: you need infinite power! As you pull horizontally more and more, less and less of this force component is available to lift the weight vertically as the two forces get closer to being at right angles (orthogonal) to each other. This is the classic problem with the asymptote: you can get closer and closer, but you can never get there.

In fact, as you get closer to the desired true horizontal angle, each unit of force at the angle of the rope gives less and less force in the vertical direction. In short: the force vector has no effect at 90⁰ on its line of action, but you can get as close as you want, even if you can’t get there completely.

If you think that neglecting the weight of the rope is a “scam” to distort or mislead the discussion, no, it is not. The use of the weight of the rope with a uniform load per unit length, towing on it, with or without a suspended weight, is the classic problem of the “catenary” and was solved hundreds of years ago (Figure 3). However, a complete solution requires the use of hyperbolic trigonometric functions (sinh and cosh) rather than the more common sines and cosines. Simplification is more informative to illuminate the basic orthogonal reality.

Figure 3: Neglecting the weight of the rope or cable itself greatly simplifies the first-order analysis; if you switch on this weight, hyperbolic trigonometric functions are required and the result is the classic catenary curve. (Image: ResearchGate)

The separation of force in its orthogonal components shows why the sails cannot sail directly in the wind, but can sail against the wind, catching at an angle. When a boat is heading directly into the wind, there is no force component in the direction the sailboat is heading.

However, if the boat is heading at an angle to the wind, then the wind force can be divided into two components (Figure 4), and the component of the force in the direction parallel to the course of the sailboat will push the boat at an angle to the wind. Usually, when moving against the wind, the sailboat will move in a cycle of about ± 45 degrees with respect to the wind direction (other factors such as water current affect the actual preferred setting). Keep in mind that the sail is usually not aligned with the hull of the sailboat and the angle between the two affects the resolution of forces.

Figure 4: The separation of forces in orthogonal comments helps to visualize how a sail can rotate and thus sail against the wind. (Image: Physics classroom)

Part 2 addresses the issue of orthogonality in terms of its formal definition and its application to communications and signal theory.

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Additional references

Orthogonality

Orthogonal versus uncorrelated: problems

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What is “orthogonal”? (Part 1): mechanical design 

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