The exact surface in this book used hard-margin, linearly separable data on purpose. This bridge names the case it deferred: nonseparable data, soft margins, slack penalties, and the constrained optimization problem that chooses the separator.
highlighted = computed this step
The hard-margin choice
This book used hard-margin data by design: the displayed points separate with every margin check satisfied. That made the geometry exact and inspectable, but it was a modeling choice. It does not say real datasets separate cleanly.
hard margin assumes separable shown data
The soft-margin name
When the data is not separable, the honest next name is soft margin. A soft-margin SVM allows margin violations and penalizes them with slack terms. Those penalties keep the violation size visible instead of pretending the hard-margin constraints still hold.
soft margin=allow slack and penalize it
Where it belongs
Choosing the soft-margin separator is a constrained quadratic program. That optimization problem is the optimization track's territory, not something this geometry-only lesson silently solves. Hard margin was the exact toy surface here; soft margin and regularization are deferred to constrained optimization.
soft margin is deferred to constrained optimization