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# Normal|Definition & Meaning

## Definition

At every given **position** along the curve, a **normal** is a **straight** line that cuts **through** the curve at a **certain point** and runs in a **direction** that is **perpendicular** to the tangent at a **certain** point. If the slope of the **line** is given by the variable n, as well as the **slope** of a tangent at **around** that point, or the value of a **gradient** or **derivative** at that point, is given by the **variable** m, then we have m **times** n equals -1.

A **normal** is a line, ray, as well as vector which is **perpendicular** to some other object in geometry. For **instance,** the **infinite**Â line **perpendicular** towards the **tangent** line to a curve at a **particular** location is the **normal** line to the curve in the **plane** at that point.

**Algebraic** signs on a normal vector’s **components** can denote the number of **sides** an object has, and its **length** can describe the object’s curvature.

A **vector** that is perpendicular to a **tangent** plane of a **surface** at a **given location** in three **dimensions constitutes** a **surface normal,** also **known** as a **normal,** to that **surface** at that point.Â

A line that is **normal** to **such** a plane and many other things are all **examples** of adjective uses of the word **“normal.”** The idea of **orthogonality** can be generalized from the **principle** of **normality.**

## What Are Normal and Tangent in Mathematics?

### Tangent

If we are **driving** a car **around** a curve, when we **drive** over **anything** slick on the road (such as oil, ice, **water,** or loose gravel), and our **automobile** starts to slide, this will **continue** in such a **direction** that is tangent towards the curve.

If we **grasp** a ball in one hand and **swing** it through a **circular motion,** then let go of it, the ball **would fly** off in a **direction** that is perpendicular to the **motion** that it was **being performed.** The **spokes** of a **bicycle** wheel are **parallel** to the rim. A bicycle wheel’s **spokes** should be **aligned perpendicular** to the rim.

### Normal

**When** driving **quickly around** with a **circular** path in an **automobile,** the force which you sense **pushing** you outwards goes **normal** to the **curvature** of the road. This is the case even if the **track** is not circular.

It is **interesting** to note that the **force** that’s also **causing** you to travel around at that corner is directed **specifically** toward the center of the circle, and it is **acting** in a **direction** that is **normal towards** the circle. At every point where a **spoke** joins with the **center** of the wheel, it is **positioned** so that it is normal towards the **circle** form of the **wheel.**

The **lines** that are linked with curves like **circles,** parabolas, **ellipses,** and hyperbolas are **referred** to as **tangents** and **normal.**

A line is said to be tangent to a curve when it touches the **curve** at exactly one point, and this particular point is referred to as the contact point. At the point of **contact,** the **normal** is a line that runs in a direction that is **perpendicular** to the tangent.

**Tangents** can be created to a curve in a variety of different ways, one for each of the individual points that make up the curve. **Because** both the **tangents** and the normal are composed of **straight lines,** their mathematical **representation** is that of a linear equation **between** x and y.

The equation for the tangent and the normal can be written in a general form as ax + **by** + c = 0. Both the equation of a tangent as well as the **equation** of a **curve** can be solved using the **contact** point as a reference point.

**Normal** and **tangents** are the lines that **define curves.** Each of the **curve’s** points has a **tangent,** which is just a line that touches the **curve** at that **point.** The **normal** is the line that cuts **through** the contact point **perpendicular** to the tangent.

## Why Do We Need Normal and Tangent?

When we are **analyzing** the forces that are **operating** on a **moving** body, we **frequently** need to locate the tangents and **normal** of the curves. A line is said to be tangent **toward** a curve if it meets the curve at **exactly** one place and if the slope of the line at that **point** is the same as the slope of the curve whereas a line that is **perpendicular** toward a **tangent** to a curve is **referred** to as the curve’s normal.

## Uses of the Surface Normal

**Surface** normals are **important** for defining vector field surface integrals.

They are also frequently utilized in **3D computer** graphics for the **purpose** of **lighting computations,** and normal mapping is frequently employed to make **necessary adjustments.**

## Tangent and Normal Properties

**Understanding** tangents and **normal** are made easier by considering their respective qualities.

**Normal**and**tangents**are**parallel**to one another.- A
**tangent’s**and a**normal’s combined**slopes are**equal**to -1. **Normal**are**inside**the curve,**while**tangents are**outside**it.- There is a
**normal**for each tangent of the curve. - The
**focus**or center of the**curve**may not always be**crossed**by the normal curve. **Normal**and tangents are both**straight**lines that are modeled by linear equations.- A
**curve**can have an**endless number**of tangents traced to it.

## What Is the Connection Between Tangents and the Normal?

The **tangents** and the **normal** are two names for the **lines** that intersect a circle at this point. While the **tangent** at that location is **parallel** to the **circle,** the normal direction of motion is **perpendicular** towards the curve. There is a gradient equal to m along both the **tangent** and the **normal** if the slope of the **curve** is m.

Two **lines** that link **curves** are **tangents** and normal. Each point on the curve has a tangent or even a path that **crosses** the **circular** at a specific location. Only at the touch point, a **normal** line is **perpendicular** to the tangent. A horizontal line that **intersects** the curve up until a given point is the **tangent** of such a curve.

Only the precise area that was previously **indicated** is touched; the curve is not **crossed.** Such a tangent has a **horizontal** line **perpendicular** to it.

## A Numerical Example of Calculating a Normal

### Example

Determine the **equation** of the **normal** to the **circle** using the following:

\[2x^2 + 2y^2 -2x -5y +3 = 0 \text{ at }(1,1)\]

### Solution

We know that:

The **circle center** is:

\[\left(\frac{1}{2},\frac{5}{4}\right)\]

Then, the **equation** to the **normal** circle passing through the given point is:

x + 2y = 3

*All mathematical drawings and images were created with GeoGebra.*