Fundamentals on Space Curve

Introduction

In differential geometry of curves in space, three vectors are called fundamental vectors. These three vectors are

1. tangent vector
2. principal normal vector, and
3. binormal vector

What is tangent Vector?

1. In a point \(P\) of a space curve, which of the following is correct?
Select Tangent has one‑point contact at \(P\) There is unique tangent at \(P\) There is infinite tangent at \(P\) Lines that pass through \(P\) and touch the curve exactly once
2. In a point \(P\) of a space curve, which of the following is NOT correct?
Select Tangent has two‑point contact at \(P\) There is unique tangent at \(P\) Tangent at \(P\) is limiting form of secant line Tangent has one‑point contact at \(P\)

What is Normal Vector

Drag the slider, check/uncheck the boxed, explore the graph and answer the questions given below.

1. Which of the following best defines a normal line?
Select Parallel to tangent Perpendicular to tangent Only the line lying in the osculating plane A line perpendicular to the osculating plane
2. How many normal lines exist at a given point?
Select There is only one unique normal line There are exactly two (Principal and Binormal) There are an infinite number of normal lines There are no normal lines

What is Principal Normal and Binormal ?

What is Principal Normal?

What is Binormal?

1. The principal normal at a point \(P\) is a line that is:
Select Parallel to the tangent Perpendicular to tangent and lies in osculating plane Perpendicular to tangent Perpendicular to tangent and osculating plane
2. The binormal at a point \(P\) is a line that is:
Select Parallel to the tangent Perpendicular to tangent and lies in osculating plane Perpendicular to tangent Perpendicular to tangent and osculating plane

Fundamental vector

What is the equation of tangent vector?
\(R=\vec{r}+\lambda \vec{t}\)
\((R-\vec{r}) \cdot \vec{t}\)
\(R=\vec{r}+\lambda \vec{n}\)
\((R-\vec{r}) \cdot \vec{n}\)
\(R=\vec{r}+\lambda \vec{b}\)
\((R-\vec{r}) \cdot \vec{b}\)

---

---

---

What is the equation of principal normal vector?
\(R=\vec{r}+\lambda \vec{t}\)
\((R-\vec{r}) \cdot \vec{t}\)
\(R=\vec{r}+\lambda \vec{n}\)
\((R-\vec{r}) \cdot \vec{n}\)
\(R=\vec{r}+\lambda \vec{b}\)
\((R-\vec{r}) \cdot \vec{b}\)

---

---

---

What is the equation of binormal vector?
\(R=\vec{r}+\lambda \vec{t}\)
\((R-\vec{r}) \cdot \vec{t}\)
\(R=\vec{r}+\lambda \vec{n}\)
\((R-\vec{r}) \cdot \vec{n}\)
\(R=\vec{r}+\lambda \vec{b}\)
\((R-\vec{r}) \cdot \vec{b}\)

---

---

---

Fundamental plane

Osculating plane

1. What is the equation of osculating plane?
\(R=\vec{r}+\lambda \vec{t}\) \((R-\vec{r}) \cdot \vec{t}\) \(R=\vec{r}+\lambda \vec{n}\) \((R-\vec{r}) \cdot \vec{n}\) \(R=\vec{r}+\lambda \vec{b}\) \((R-\vec{r}) \cdot \vec{b}\)
2. Which vector span the osculating plane?
\( \vec{t}\) and \(\vec{n}\) \(\vec{t}\) and \(\vec{b}\) \(\vec{b}\) and \(\vec{n}\)
3. Which vector is perpendicular to the osculating plane?
\(\vec{t}\) \(\vec{n}\) \(\vec{b}\)

---

---

---

Normal plane

4. What is the equation of normal plane?
\(R=\vec{r}+\lambda \vec{t}\) \((R-\vec{r}) \cdot \vec{t}\) \(R=\vec{r}+\lambda \vec{n}\) \((R-\vec{r}) \cdot \vec{n}\) \(R=\vec{r}+\lambda \vec{b}\) \((R-\vec{r}) \cdot \vec{b}\)
5. Which vector span the normal plane?
\( \vec{t}\) and \(\vec{n}\) \(\vec{t}\) and \(\vec{b}\) \(\vec{b}\) and \(\vec{n}\)
6. Which vector is perpendicular to the normal plane?
\(\vec{t}\) \(\vec{n}\) \(\vec{b}\)

---

---

---

Rectifying plane

7. What is the equation of rectifying plane?
\(R=\vec{r}+\lambda \vec{t}\) \((R-\vec{r}) \cdot \vec{t}\) \(R=\vec{r}+\lambda \vec{n}\) \((R-\vec{r}) \cdot \vec{n}\) \(R=\vec{r}+\lambda \vec{b}\) \((R-\vec{r}) \cdot \vec{b}\)
8. Which vector span the rectifying plane?
\( \vec{t}\) and \(\vec{n}\) \(\vec{t}\) and \(\vec{b}\) \(\vec{b}\) and \(\vec{n}\)
9. Which vector is perpendicular to the rectifying plane?
\(\vec{t}\) \(\vec{n}\) \(\vec{b}\)