The Internet is enticing a rapidly growing number of shoppers to make a very personal purchase, prescription eyeglasses, online. Deep discounts and greater variety are prompting many to try something new.

Customers can’t pluck a pair of glasses from their smartphone screen to learn how they feel, but shoppers can try on frames virtually or have them delivered for a free test. They also can quickly scroll through hundreds of choices and send pictures to friends for a second-opinion.