Farid Tari is a Full Professor at the University of São Paulo. He has published over 60 research papers, including some in leading mathematical journals, authored one book, and edited another. He has supervised several postdoctoral fellows, PhD, and MSc students. Currently, he serves as an Associate Editor of the Journal of Singularities.Mostafa Salarinoghabi is currently a postdoctoral researcher at the Federal University of Viçosa, Brazil. He earned his PhD from the University of São Paulo, Brazil. His expertise lies in differential geometry, singularity theory, and random complex dynamics.Masaru Hasegawa is currently an Assistant Professor at Iwate Medical University, Japan. He earned his PhD from Saitama University, Japan, where he also held a postdoctoral position. Additionally, he completed another postdoctoral position at the University of São Paulo, Brazil. His research focuses on applying singularity theory to differential geometry.

• Chapter 1. Map-germs from the plane to the plane.- Chapter 2. Geometric deformations of discriminants.- Chapter 3. Geometric deformations of the fold and cusp.- Chapter 4. Ae-codimension 1 singularities.- Chapter 5. Ae-codimension 2 singularities.- Chapter 6. Apparent contours.- Chapter 7. Geometric invariants.

Farid Tari is a Full Professor at the University of São Paulo. He has published over 60 research papers, including some in leading mathematical journals, authored one book, and edited another. He has supervised several postdoctoral fellows, PhD, and MSc students. Currently, he serves as an Associate Editor of the Journal of Singularities.Mostafa Salarinoghabi is currently a postdoctoral researcher at the Federal University of Viçosa, Brazil. He earned his PhD from the University of São Paulo, Brazil. His expertise lies in differential geometry, singularity theory, and random complex dynamics.Masaru Hasegawa is currently an Assistant Professor at Iwate Medical University, Japan. He earned his PhD from Saitama University, Japan, where he also held a postdoctoral position. Additionally, he completed another postdoctoral position at the University of São Paulo, Brazil. His research focuses on applying singularity theory to differential geometry.

• Chapter 1. Map-germs from the plane to the plane.- Chapter 2. Geometric deformations of discriminants.- Chapter 3. Geometric deformations of the fold and cusp.- Chapter 4. Ae-codimension 1 singularities.- Chapter 5. Ae-codimension 2 singularities.- Chapter 6. Apparent contours.- Chapter 7. Geometric invariants.

Farid Tari is a Full Professor at the University of São Paulo. He has published over 60 research papers, including some in leading mathematical journals, authored one book, and edited another. He has supervised several postdoctoral fellows, PhD, and MSc students. Currently, he serves as an Associate Editor of the Journal of Singularities.Mostafa Salarinoghabi is currently a postdoctoral researcher at the Federal University of Viçosa, Brazil. He earned his PhD from the University of São Paulo, Brazil. His expertise lies in differential geometry, singularity theory, and random complex dynamics.Masaru Hasegawa is currently an Assistant Professor at Iwate Medical University, Japan. He earned his PhD from Saitama University, Japan, where he also held a postdoctoral position. Additionally, he completed another postdoctoral position at the University of São Paulo, Brazil. His research focuses on applying singularity theory to differential geometry.

• Chapter 1. Map-germs from the plane to the plane.- Chapter 2. Geometric deformations of discriminants.- Chapter 3. Geometric deformations of the fold and cusp.- Chapter 4. Ae-codimension 1 singularities.- Chapter 5. Ae-codimension 2 singularities.- Chapter 6. Apparent contours.- Chapter 7. Geometric invariants.